LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class SELF-TAUGHT MECHANICAL DRAWING AND ELEMENTARY MACHINE DESIGN A Treatise Comprising the First Principles of Geometric and Mechanical Drawing, Workshop Mathematics, Mechanics, Strength of Materials, and the Design of Machine Details, including Cams, Sprockets, Gearing, Shafts, Pulleys, Belting, Couplings, Screws and Bolts, Clutches, Flywheels, etc. Pre- pared for the Use of Practical Mechanics and Young Draftsmen. By F. L. .SYLVESTER, M.E. With Additions By ERIK OBERG Associate Editor of "Machinery," Author of "Hand-Book of Small Tools," "Shop Arithmetic for the Machinist," "Advanced Sho^ Arithmetic for the Machinist " "The Use of Logarithms," ''Solution of Triangles," etc. FULLY ILLUSTRATED NEW YORK THE NORMAN W. HENLEY PUBLISHING CO. 132 NASSAU STREET 1910 < eA Copyrighted, 1910, by The Norman W. Henley Publishing Co. PREFACE THE demand for an elementary treatise on mechanical drawing, including the first principles of machine design, and presented in such a way as to meet, in particular, the needs of the student whose previous theoretical knowledge is limited, has caused the author to prepare the present vol- ume. It has been the author's aim to adapt this treatise to the requirements of the practical me- chanic and young draftsman, and to present the matter in as clear and concise a manner as possible, so as to make " self -study " easy. In order to meet the demands of this class of students, practically all the important elements of machine design have been dealt with, and, besides, algebraic formulas have been explained and the elements of trigo- nometry have been treated in a manner suited to the needs of the practical man. In arranging the material, the author has first devoted himself to mechanical drawing, pure and simple, because a thorough understanding of the principles of representing objects greatly facilitates further study of mechanical subjects ; then, atten- tion has been given to the mathematics necessary iii 01 9.J.4.1 IV PREFACE for the solution of the problems in machine design presented later, and to a practical introduction to theoretical mechanics and strength of materials; and, finally, the various elements entering in ma- chine design, such as cams, gears, sprocket wheels, cone pulleys, bolts, screws, couplings, clutches, shafting, fly-wheels, etc., have been treated. This arrangement makes it possible to present a con- tinuous course of study which is easily compre- hended and assimilated even by students of limited previous training. Portions of the section on mechanical drawing was published by the author in The Patternmaker several years ago. These articles have, however, been carefully revised to harmonize with the pres- ent treatise, and in some sections amplified. In the preparation of the material, the author has also consulted the works of various authors on machine design, and credit has been- given in the text wherever use has been made of material from such sources. Several important additions have been made by Mr. Erik Oberg, Associate Editor of Machinery. In the preparation of these additions, use has partly been made of material published from time to time in Machinery. THE PUBLISHER. APRIL, 1910. CONTENTS PREFACE Page iii CHAPTER I INSTRUMENTS AND MATERIALS General Remarks on the Study of Drawing Drawing Instruments Pencils Use of the Instruments Paper Ink Page 1 CHAPTER II DEFINITIONS OF TERMS USED IN GEOMETRICAL AND MECHANICAL DRAWING Point Line Surface Solid Plane Angle Circle Parallelogram Polygon Ellipse Involute Cycloid Parabola Page 10 CHAPTER III GEOMETRICAL PROBLEMS Bisecting of Lines and Angles Perpendicular Lines Tangents Regular Polygons Inscribed and Cir- cumscribed Circles Ellipses Spirals Involutes Cycloids Parabolas Page 17 Vi CONTENTS CHAPTER IV PROJECTION Mode of Representing Objects Projections of Inclined Prisms Surface Developments of Cones and Pyra- mids Intersecting Cylinders, and Cylinder and Cone Projection of a Helix Isometric Projec- tion Page 32 CHAPTER V WORKING DRAWINGS Object of Working Drawings Assembly Drawings Detail Drawings Dimensions Finish Marks Sectional Views Cross-section Chart Screw Threads Shade Lines Tracing and Blue-print- ing Page 50 CHAPTER VI ALGEBRAIC FORMULAS The Meaning of Formulas Square and S'quare Root Cube and Cube Roots Exponents Areas and Volumes of Plane Figures and Solids Page 79 CHAPTER VII ELEMENTS OF TRIGONOMETRY Angles Right-angled Triangles Trigonometrical Functions Tables of Natural Functions Solution of Right-angled Triangles Solution of Oblique- angled Triangles Laying Out Angles by Means of Trigonometric Functions Page 96 CONTENTS Vii CHAPTER VIII ELEMENTS OF MECHANICS Resolution of ForcesLevers Fixed and Movable Pulleys Inclined Planes The Screw Differential Screw Newton's Laws of Motion Pendulum Falling Bodies Energy and Work Horse-power of Steam Engines Page 120 CHAPTER IX FIRST PRINCIPLES OF STRENGTH OF MATERIALS Factor of Safety Shape of Machine Parts Strength of Materials as Given by Kirkaldy's Tests Stresses in Castings Page 151 CHAPTER X CAMS General Principles Design of Cams Imparting Uniform Motion Reciprocating Cams Cams Providing Uniform Return Uniformly Accelerated Motion Cams Gravity Cam Curve Harmonic Action Cams Approximate Gravity Cam Curve . .Page 164 CHAPTER XI SPROCKET WHEELS Object of Sprocket Wheels Drafting of Sprocket Wheels for Different Classes of Chain Speed Ratio Page 185 Viii CONTENTS CHAPTER XII GENERAL PRINCIPLES OF GEARING Friction and Knuckle Gearing Epicycloidal Gearing Gears with Strengthened Flanks Gears with Radial Flanks Involute Gears Interference in Involute Gears The Two Systems Compared Twenty-degree Involute Gears Shrouded Gears Bevel Gears Worm Gearing Circular Pitch Proportions of Teeth Diametral Pitch The Hunting Tooth Approximate Shapes for Cycloidal Gear Teeth Involute Teeth Proportions of Gears Strength of Gear Teeth Thurston's Rule for Gear Shafts Speed Ratio of Gearing. . .Page 190 CHAPTER XIII CALCULATING THE DIMENSIONS OF GEARS Spur Gearing Bevel Gears Worm Gearing. . Page 222 CHAPTER XIV CONE PULLEYS Conical Drums Influence of Crossed Belt Cone Pulleys Smith's Rule for Laying Out Cone Pulleys Page 239 CHAPTER XV BOLTS, STUDS AND SCREWS Kinds of Screws United States Standard Screw Thread Check or Lock Nuts Bolts to Withstand Shock Wrench Action Screws for Power Transmission Efficiency of Screws Acme Standard Thread- Miscellaneous Screw Thread Systems Other Com- mercial Forms of Screws Page 243 CONTENTS ix CHAPTER XVI COUPLINGS AND CLUTCHES Simple Forms of Couplings Calculation of Flange Coupling Bolts Oldham's Coupling Hooke's Coupling or Universal Joint Toothed Clutches Friction Clutches Cone Clutches Page 259 CHAPTER XVII SHAFTS, BELTS AND PULLEYS Calculation of Shafting Horse-power of Belting Speed of Belting Pulley Sizes and Speed Ratios Twisted and Unusual Cases of Belting . .Page 272 CHAPTER XVIII FLY-WHEELS FOR PRESSES, PUNCHES, ETC. Object of Fly-wheels Formulas for Fly-wheel Calcu- lations Example of Fly-wheel Calculation for Shears Page 289 CHAPTER XIX TRAINS OF MECHANISM To Secure Increase of Speed To Secure Reversal of Direction The Compound Idler The Screw Cut- ting Train Simplified Rules for Calculating Lathe Change Gears Back-Gears Page 297 X CONTENTS CHAPTER XX QUICK RETURN MOTIONS Object of Quick Return Motions Examples of Simple Designs of Quick Return Motions The Whitworth Quick Return Device The Elliptic Gear Quick Return Page 313 SELF-TAUGHT MECHANICAL DRAWING CHAPTER I INSTRUMENTS AND MATERIALS ONE who is to study the subject of drawing should not merely read a book on the subject, but should prepare sheets of exercises. This will fix the principles which he learns in his mind in a way as reading alone will not do, and will give him practical experience in the use of the tools. The geometrical problems given in this book make perhaps the best of subjects for a beginning, as their proper execution will require careful work. Later, the student may make dimensioned free- hand sketches of some machine with which he is familiar, and from these sketches he may make up a set of finished working drawings. In all of this work, care should be taken to have it so laid out, with proper margins and spaces between different parts, that the drawing when finished shall pre- sent an appearance of neatness and methodical arrangement. For the purposes of the student, a drawing board about 15 by 18 inches will be large enough. With this should be an 18-inch T-square, a pair of 6-inch triangles, and a set of three or four irregular curves. 1 SELF-TAUGHT MECHANICAL DRAWING For drawing full-size work, a good flat beveled- edge rule will answer ordinary requirements, but for making half- or quarter-size drawings some kind of a "scale" ' will be found desirable. The tri- angular scale shown in Fig. 1 is perhaps the one mostly used, and it has the advantage of possess- \ Ytt \V\\ \ Y\\ \\\\ \\\\ \X\\ \\Tvl\\ \Y\\ \ FIG. 1. -The Triangular Scale. ing six surfaces for graduations, giving variety enough for all sorts of conditions, but it has the disadvantage of persistently presenting the wrong edge, and putting one to the trouble of turning it over and over to get the desired edge. This trouble may, of course, be overcome by using a scale guard such as is shown in Fig. 2, but the guard is itself often in the way. As but two or three differ- ent scales, aside from full size, will be likely to be required, it will be found much more con- FIG. 2. -Scale Guard or Holder venient to have a sep- used on Triangular Scale. arate flat scale for each graduation. Such scales may be purchased, or, if one is satisfied with the open graduation system shown in Fig. 3, he may make them without much trouble himself. In this system, only one inch is divided, this inch being numbered 0; and measurements which include a INSTRUMENTS AND MATERIALS 3 fractional part of an inch are reckoned from the required whole number to the proper place on the divided inch. The drawing instruments themselves, while not necessarily of the highest price, should be of a good serviceable quality of German silver. The cheap brass or nickel plated school sets should not be considered, as they will prove unsatisfactory for regular work. It is not necessary to have a large number of instruments. A very good set, sufficient for all ordinary requirements, might be as follows: First a pair of about 4J- or 5-inch com- 11 12 13 14 15 16 17 "18 FIG. 3. Inexpensive Type of Scale. passes with fixed needle points (bayonet points are useless) and interchangeable pin and pencil points, with lengthening bar. Then, a pair of hair- spring spacers of about the same size. These re- semble ordinary plain compasses, but the steel end of one leg is made adjustable by means of a thumb screw. Next, a pair of ruling pens, one large and one small, and, lastly, a set of three spring instruments, pen, pencil and spacers, for small work. Rather than to get cheap instru- ments, it would be advisable to obtain a set gradu- ally by getting the large instruments and one pen first, and adding the second pen and the spring instruments later. The large compasses can, if necessary, be used to make circles of from about i inch to about 18 or 20 inches in diameter, so 4 SELF-TAUGHT MECHANICAL DRAWING that they will do very well for a beginning. For making larger circles, beam compasses, in which separate heads for the needle point and for the pen or pencil point are attached to a wooden bar, after the manner of workmen's trammels, are used. A convenient case for the instruments, when they are bought separately, is shown in Fig. 4, and is made as follows: Take two pieces of chamois skin or thin broadcloth, one of them about one-half longer than the longest instrument, and somewhat wider than all of them when they are FIG. 4. Home-made Instrument Case. laid out side by side, and the second one of the same width as the first, but somewhat shorter than the longest instrument. This second piece is sewed onto the large piece at one end by the outer edges. Pockets for the reception of the instru- ments are then made as shown, and when the free end of the large piece is folded over, the instru- ments are rolled up together. The pencils, which to avoid scratching particles, should be of best quality, should not be sharpened to a round point, but to a flat oval point, as such a shape will wear longer than a round point ; the leads used in the compasses, however, should be INSTRUMENTS AND MATERIALS 5 only slightly flattened. It will be found desirable to have two grades of pencils, one quite hard, about "4H," to be used for laying out work, and a softer one, about "2H," to be used for going over the lines of work which is not to be inked in. In laying-out work where the hard pencil is used, only a moderate pressure should be applied, so as to permit of erasures at any time, whether for the purpose of making alterations, or to free the draw- ing of pencil marks after inking. The drawing pens should be kept sharp, though not so sharp as to cut the paper, and their ends should present a neat oval shape. The needle points of the compasses should also be kept sharp to avoid the tendency to slip when doing work where it is undesirable to prick through the pa- per. A small Arkansas stone will be found useful for this purpose. Where much use is made of a given center, it may be desirable to employ a horn or metal center, such as are kept in stock by deal- ers in artists' supplies, to avoid the troublesome enlargement of the center in the paper which the points of the compasses would otherwise make. In making a drawing, care should be taken to have the preliminary pencil work done correctly. It is a mistake which beginners are likely to make, to think that errors in the pencil work may be readily corrected in the inking. This, however, is usually another case where "haste makes waste." It is much better to spend a little extra time on the pencil work, than to have to throw away a nearly finished ink drawing and do the work all over again. In locating the various 6 SELF-TAUGHT MECHANICAL DRAWING views of a drawing upon the paper, it will fre- quently be found to be well to make rough sketches of it on scrap paper. These sketches can then be moved around on the drawing paper until the best arrangement is secured. In making a drawing, it will be found most con- venient, ordinarily, to limit the use of the T-square to horizontal lines, the head of the square being kept pressed firmly against the left-hand end of the drawing board. Vertical lines are then made FIG. 5. Appearance of Carelessly made Drawing. with the aid of the triangles resting against the blade of the T-square. Vertical lines which are too long to be made in this way, are, of course, made with the T-square itself. In inking in a drawing, it is best to draw all curved or circular lines first, as it is easier to join straight lines onto curved lines than to join curved lines onto straight lines. Care should also be taken to have meeting lines just meet, whether they meet end to end or at an angle. Carelessness in this respect gives a drawing a very bad appearance, as shown by Fig. 5, A and B. INSTRUMENTS AND MATERIALS 7 In using the pens, whether the ruling or the com- pass pens, care should be taken to see that both nibs rest upon the paper, otherwise lines such as shown in Fig. 6 may result. If the pen does rest squarely upon the paper, and such lines continue to appear, it is fair to infer that the paper has become some- what greasy, perhaps from too much handling. This trouble may be avoided, and the work kept cleaner, by having a piece of thin paper inter- posed between the hands and the drawing paper. The cross hatching work, such as is shown at A in Fig. 5, is frequently done by simply using one of the triangles resting against the blade of the FlG. 6. Line Resulting from not Having both Pen Points or Nibs Resting on the Paper when Inking. T-square, the same as is done for vertical lines, the spacing being done entirely by the eye; but unless one is doing a good deal of this work, so as to keep in practice, he will find it very difficult to make the spacing regular. There are various sec- tion-lining devices on the market for doing this work, some of them quite expensive. Fig. 7 shows a simple device for cross-sectioning, which serves the purpose as well as any of the more elaborate ones, and possesses the additional advantage that anyone may readily make it for himself. This instrument was shown by Mr. E. W. Beardsley in Machinery, September, 1905. An old instrument screw, B, is screwed into a slightly smaller hole in a piece of wood, A, shaped as shown, and of a 8 SELF-TAUGHT MECHANICAL DRAWING thickness a little in excess of the diameter of .the screw-head. This combination is then used in the central hole in a triangle, as shown. Then, with one finger on the triangle itself, and with another one on A, the two may be moved along, first one and then the other, for section lining, the desired width of space being secured by the adjustment given to B. For making erasures of ink lines on paper, a steel scraping eraser or a sharp knife blade is usu- FIG. 7. Simple Cross-section Liner. ally the best, the roughened surface being after- wards rubbed down smooth with some hard sub- stance. When making erasures of either pencil or ink with a rubber eraser, an erasing shield, such as is shown in Fig. 8, is useful for prevent- ing rubbing out more than is intended. These shields are made both of thin sheet metal and of celluloid; the metal ones, being the thinner, are the more convenient to use. Tho paper used, if good work is desired, should INSTRUMENTS AND MATERIALS 9 be regular drawing paper, whether it be white or brown. This has an unglazed surface, and will be found much more satisfactory in every way than common paper. The glazed surface of the cheaper paper does not take pencil marks well, and is torn up badly in making erasures. Such paper, if used at all, should be used only on the most temporary FIG. 8. Erasing Shield made from Sheet Metal or Celluloid. work. Of white drawing papers, the smooth sur- faced kinds should be selected. For making ink drawings, it will be found most satisfactory to use the prepared drawing inks, rather than to go to the trouble of preparing it oneself from the stick India ink. For fastening the paper on to the board, common one-half-ounce copper tacks are as good, if not preferable, to other fastening means. CHAPTER II DEFINITIONS OF TERMS USED IN GEOMETRICAL AND MECHANICAL DRAWING 1. A Point has position, but not magnitude. 2. A Line has length, but neither breadth nor thickness. 3. A Surface has length and breadth, but not thickness. 4. A Solid has length, breadth and thickness. 5. A Plane is a surface which is straight in every direction; that is, one which is perfectly flat. 6. Parallel lines are such as are everywhere equally distant from each other. Circular lines which answer to this condition are also said to be concentric. 7. An Angle is the difference in the direction of two lines. If the lines meet, the point of meet- ing is called the vertex of the angle, and the lines ab and ac, Fig. 9, are its sides. 8. If a straight line meets another so that the adjacent angles are equal, each of these angles is a right angle, and the two lines are perpendicular to each other. Thus the angles acd and deb, Fig. 10, are right angles, and the lines ab and dc are perpendicular to each other. A distinction is to be made here between the words perpendicular 10 DEFINITIONS OF TERMS 11 and vertical. A vertical line is one which is per- pendicular to the plane of the earth's horizon ; that is, to the surface of still water. 9. An Obtuse Angle is one which is greater than a right angle, as ace, Fig. 10. 10. An Acute Angle is one which is less than a right angle, as ecb, Fig. 10. 11. It is obvious that the sum of all the angles which may be formed about the point c, Fig. 10, above the line ab will be equal to the two right angles acd and deb. FIG. 9. Angle. FIG. 10. Illustration for Making Clear the Terms Right, Acute and Obtuse Angles. 12. The Complement of an angle is a right angle, less the given angle. Thus bce t Fig. 10, is the complement of dee. 13. The Supplement of an angle is two right angles less the given angle. Thus bee, Fig. 10, is the supplement of ace. 14. A Circle is a continuous curved line, Fig. 11, or the space enclosed by such line, every point of which is equally distant from a point within called the center. 15. The distance across a circle, measured through the center, is the diameter. The distance around the circle is the circumference. The dis- 12 SELF-TAUGHT MECHANICAL DRAWING tance from the center to the circumference is the radius. 16. The ratio between the circumference and the diameter, that is, the circumference divided by the diameter, is 3.1416. While this is not exact (Bradbury's Geometry states that it has been car- ried out to two hundred and fifty places of deci- mals), it is near enough for practical purposes. This ratio is frequently represented by the Greek letter TT (pi). 17. A circle is considered as being equally divided FIG. 11. -Illustration for FIG. 12. -Similar Triangles. Making Clear the Terms Relating to the Circle. into three hundred and sixty degrees (360), each degree into sixty minutes (60'), and each minute into sixty seconds (60"). 18. If two diameters cross each other at right angles, the circle is divided into four equal parts; hence a right angle contains ninety degrees. 19. An Arc of a circle is any part of its circum- ference, as abc, Fig. 11. 20. A Chord is a straight line joining the ends of an arc, as ac, Fig. 11. 21. Two triangles, as abc and dec, Fig. 12, hav- ing like angles are similar triangles. The corre- DEFINITIONS OF TERMS 13 spending sides of similar triangles have the same ratio. Thus if ac were twice as long as dc, ab would be twice as long as de, and be would be twice as long as ec. 22. The sum of the angles of a triangle is equal to two right angles. Let abc, Fig. 13, represent any triangle. Extend one side, ac, as shown, and make cd parallel with ab. Then the angle dee is equal to the angle bac, for their sides have the same direction, and the angle bed is equal to the FIG. 13. Illustration for Showing that the Sum of the Angles in a Triangle equals Two Right Angles. FIG. 14. Tangent and Nor- mal to a Curve. angle abc, for their sides have opposite directions; hence the sum of the three angles formed about the point c is equal to the sum of the three angles of the triangle abc, and these are equal to two right angles (11). 23. A Tangent is a line which touches another, but does not, though extended, cross it. Thus, a, b and c, Fig. 14, are tangent lines. A line, d, perpendicular to the straight line 6, at the point of tangency, is called a normal. If one of the 14 SELF-TAUGHT MECHANICAL DRAWING lines, as a, is circular, the normal will pass through its center. 24. A Parallelogram is a figure whose opposite sides are parallel, as ab and cd, or eb and fd in Fig. 15. The sides may all be of equal length, FIG. 15. Parallelograms. FIG. 16. -Square. (See when the parallelogram is called a square. Fig. 16.) 25. Figures having five, six or eight sides are called respectively Pentagon, Hexagon and Octagon. These, and all figures having more than four sides, are called Polygons. If the sides in a polygon are FIG. 17. Regular Polygon. FIG. 18. Ellipse. all of equal length, and all the angles equal, the polygon is called a regular polygon. (See Fig. 17.) 26. An Ellipse, Fig. 18, is a continuous curved line, or the space enclosed by such line, of such shape that the sum of the distances from two DEFINITIONS OF TERMS 15 points within, as a and 6, called the foci (singu- lar: focus), to any point upon its circumference is constant. Thus al plus bl equals a2 plus b2 or a3 plus b3. 27. An Involute is a line of such shape (as a in FIG. 19. Involute. FIG. 20. Cycloid. Fig. 19) as might be made by a pencil at the end of a string which is unwound from a circle. 28. A Cycloid is a line of such shape (as a in Fig. 20) as might be made by a pencil fastened to the circumference of a circle which is being rolled upon a straight line. If the circle was being rolled upon the convex side of a circular line the line traced by the pencil would be an epicy- cloid. If it was being rolled upon the concave side of a circular line, the line traced by the pencil would be a hypocycloid. The involute and cycloidal curves are used in gear outlines. 29. A Parabola is a curve which may be ob- FIG. 21. Method of Sec- tioning a Cone to Ob- tain a Parabola. 16 SELF-TAUGHT MECHANICAL DRAWING tained by cutting a cone so that the exposed sectional surface will be parallel with one of the sides of the cone, as shown in Fig. 21. This curve, as shown in Fig. 22, is of such shape that lines drawn to it from a certain point within, called the focus, shown at / in the illustration, niake the same angle with it as lines drawn from /7 /\7 FIG. 22. Parabola. the intersection points parallel with the axis ax. Thus the line fm makes the same angle with the parabola, at the point of intersection, as the line ml. Because of this property of the parabola, mirrors of this shape are used in headlights of locomotives, in search lights," and in many light- houses ; because, if a light be placed at the focus, its rays, when reflected from the mirror, will be thrown out in parallel lines. CHAPTER III GEOMETRICAL PROBLEMS Prob. 1, Fig. 23. To bisect a line, either curved as abc, or straight as ac. With centers at a and c and with a radius somewhat greater than half the length of the line, describe the arcs d and e. A line passing through the intersections of these arcs bisects either line. It will also pass through the center of the circle of which the arc abc is a part. Prob. 2, Fig. 24. To bisect an angle. With FIG. 23. Bisecting a Line. FIG. 24. Bisecting an Angle. center at a, and with any convenient radius, de- scribe the arc be. With centers at b and c, and with a radius greater than half the arc, describe the arcs d and e. A line from a through the inter- section of these arcs bisects the angle. Prob. 3, Fig. 25. To make an angle equal to a given angle. Let a be the given angle, and let it be desired to make an angle equal to it on the line dg. With center at a make the arc be, and then with center at d make the arc eh with the same 17 18 SELF-TAUGHT MECHANICAL DRAWING radius. Then with a radius equal to be, and with center at h, make the arc /. A line from d through the intersection of the arcs gives the required angle. Prob. 4, Fig. 26. To erect a perpendicular at the end of a line, ab. With any convenient center, c, FIG. 25. Making an Angle Equal to a Given Angle. and with radius cb, draw a semicircle intersecting ab at d. Draw a line from d through c intersect- ing the semicircle at e. A line from 6 passing through e is the required perpendicular. Prob. 5, Fig. 27. To drop a perpendicular from a point a, to a given line be. With a as a center, FIG. 26. Erecting a Perpen- dicular Line. r FIG. 27. Drawing a Perpen- dicular Line. draw an arc intersecting be at d and e. With d and e as centers draw the intersecting arcs / and g. A line from a through the intersection of these arcs is the required perpendicular. If a were over one end of the line be the process shown GEOMETRICAL PROBLEMS 19 in the preceding problem might be reversed by drawing a line from a corresponding to de, Fig. 26, and upon this line drawing a semicircle, when its intersection with the base line would give the point to which the perpendicular from a should be drawn. Prob. 6, Fig. 28. To draw a tangent to a circle at a given point. Draw a radius of the circle to the required point, and erect a perpendicular to it, which will be the required tangent. To find the point of tangency of a line to a circle, drop a per- FIG. 28. Drawing a Tangent FIG. 29. Finding the Center to a Circle. of a Circle. pendicular to the tangent from the center of the circle. Prob. 7, Fig. 29. To find the center of a circle. Mark off two arcs as ab and ac upon the circumfer- ence, and bisect these arcs as in Prob. 1. Where these bisecting lines cross each other will be the required center. Prob. 8, Fig. 30. To draw a regular hexagon upon a given base, ab. With a radius equal to the length of ab draw the arcs c and d. The intersec- tion of these arcs will be the center of a circum- scribing circle upon which the other sides may be marked off. 20 SELF-TAUGHT MECHANICAL DRAWING Prob. 9, Fig. SI. To draw a regular octagon in a square. Draw the diagonals of the square, ad and be, and with a radius equal to half of a diago- nal, and with centers at a, b, c and d, draw the arcs e, f, g and h. The intersections of these arcs FIG. 30. Drawing a Regular Hexagon. FIG. 31. Drawing a Regular Octagon. with the sides of the square give the corners of the required octagon. Prob. 10, Fig. 32. To draw a circle about a tri- angle, as abc. Bisect any two of the sides as in Prob. 1. Where the bisecting lines cross each 32. Drawing a Circle about a Triangle. FIG. 33. Inscribing a Circle in a Triangle. other will be the center of the required circle. In a similar manner a center may be found from which to draw a circle through any three given points, the given points in this case being the cor- ners of the triangle. GEOMETRICAL PROBLEMS 21 Prob. 11, Fig. 33. To draw a circle within a given triangle, as abc. Bisect any two of the angles as in Prob. 2. Where the bisecting lines cross, will be the center of the required circle. In a similar manner a center may be found from which to draw a circle tangent to any three given straight lines. Prob. 12, Fig. 34. To find the foci of an ellipse. Draw the long and the short diameters of the ellipse, ab and cd, and with a radius equal to half of the long diameter, and with a center at c or d FIG. 34. Finding the Foci of an Ellipse. FIG. 35. Simplified Method of Drawing an Ellipse. draw the arcs e and /. Where these arcs intersect the long diameter will be the required foci. Prob. 13, Fig. 35. To draw an ellipse with a pencil and thread. Having found the foci of the ellipse, stick a pin firmly into each focus, and loop- ing a thread around them, allow it to be slack enough so that the pencil will draw it out to the end of the short diameter. The thread will then guide the pencil so that it will draw an ellipse. A groove should be cut around the pencil lead to pre- vent the thread from slipping off. Prob. 14, Fig. 36. To draw an ellipse with a trammel. Lay out the long and the short diame- ters of the ellipse, ab and cd, and on a strip of paper, A, mark off 1-3 equal to half of the long diam- 22 SELF-TAUGHT MECHANICAL DRAWING eter, and 2-3 equal to half of the short diameter. Then, keeping point 1 on the short diameter, and point 2 on the long diameter, mark off any desired number of points at 3. A curved line passing through these points will be the required ellipse. The ellipsograph, an instrument for drawing el- lipses, is made on this principle, points at 1 and 2 traveling in grooves which coincide with ab and cd. Prob. 15, Fig. 37. To draw an ellipse by tangent lines. Make ab equal to one-half of the long di- d FIG. 36. Another Method of FIG. 37. Drawing an Ellipse Drawing an Ellipse. by Tangents. ameter of the required ellipse, and be equal to one- half its short diameter. Divide ab and be into the same number of equal parts, and, numbering them as indicated, connect 1 and 1' ', 2 and 2' and so forth. A curved line starting at a, tangent to these lines, and ending at c, is one-quarter of the required ellipse. Prob. 16, Fig. 38. To draw an approximate el- lipse with compasses, using four centers. Lay out the long diameter ab, and the short diameter cd, crossing each other centrally at o. From 6 meas- ure off be equal to co, one-half of the short diam- eter. The length ae will then be the radius gh for forming the part hk of the ellipse. From e GEOMETRICAL PROBLEMS 23 mark off the point/, making ef equal to one half of oe. The point / will be the center, and fb the radius for forming the end of the ellipse. Lines drawn from the centers g through the points / de- termine the points at which the different curves meet. This method is not considered applicable when the short diameter is less than two-thirds of the long diameter. FIG. 38. Drawing an Approximate Ellipse by Four Circular Arcs. Prob. 17, Figs. 39 and 39a. To draw an approx- imate ellipse with compasses, using eight centers. Lay out the long diameter ab, and the short diam- eter cd crossing each other centrally at /. Con- struct the parallelogram aecf, and draw the diago- nal ac. From e draw a line at right angles to ac, crossing the long diameter at h, and meeting the short diameter, extended, at g. Point g is the center from which to strike the sides of the ellipse, and 24 SELF-TAUGHT MECHANICAL DRAWING h will be the center, subject to certain modifica- tions for narrow ellipses, from which to strike the ends of the ellipse. To get the radius of the third curve for connecting the side and end curves, lay off a base line ab, Fig. 39 A, of any convenient length, and divide it into five equal parts by the points 1, 2, 3 and 4. At one end of the line erect the perpendicular ac, equal to the end radius ah, and at the other end erect the perpendicular bd equal to the side radius eg. Connect the ends of these perpendiculars by the line cd, and at point 2 erect a perpendicular, meeting cd at e. The length e2 will be the desired third radius. With the compasses set to this radius, find a center i from which a curve can be struck which will be just tangent to the side and end curves. From other centers similarly located the remainder of the ellipse is drawn. Lines drawn from i through h, and from g through i determine the meeting points of the different curves. For narrow ellipses the length of the end radius, ah, should be increased as follows : For an ellipse having its breadth equal to one-half of its length, make ah one-eighth longer. For an ellipse having its breadth one-third of its length, make ah one- fourth longer. For an ellipse having its breadth equal one-quarter of its length, make ah one-half longer. For intermediate breadths lengthen ah proportionately. With this modification of the length of the end radius, this method gives curves which blend well together so as to satisfy the eye, and gives a figure which conforms quite closely to the actual outlines of an ellipse. GEOMETRICAL PROBLEMS 25 FIG. 39a. FIGS. 39 and 39a. Drawing an Approximate Ellipse by Eight Circular Arcs. 26 SELF-TAUGHT MECHANICAL DRAWING Prob. 18, Fig. 40. To draw a regular polygon of any number of sides on a given base, ab. Extend ab as shown, and on it with one end as a center and a radius equal to the length of the given side, draw a semicircle. Divide this semicircle into as many equal spaces as there are to be sides to the polygon. A line from b to the second space, reckoning from where the semicircle meets the extension of ab, will be a second side of the required polygon. Lines are then drawn from b through the remain- ing divisions of the semicircle, and the remaining FIG. 40. Drawing a Regular FIG. 41. Drawing a Spiral Pentagon. about a Square. sides of the polygon are marked off upon them as indicated. If the polygon is to have many sides, as an additional precaution against error, bisect ab and b2, thus getting the center of a circumscribing circle upon which the remaining sides may be marked off. Prob. 19, Fig. 41. To draw a spiral about a square. Lay out a square, 1-2-3-4, having the length of each side equal to one-quarter of the de- sired distance between the successive convolutions of the spiral, and extend each side in one direction as shown. With a center at 2, and with a radius 1-2 draw a quarter of a circle. With a center at 3 GEOMETRICAL PROBLEMS 27 draw another quarter of a circle, continuing the first one, and so continue with successive corners of the square for centers. Fig. 42 shows how, by similarly extending one end of each side, a spiral may be drawn about a regular polygon of any number of sides. A curve so formed determines the shape of the teeth of sprocket wheels. Prob. 20, Fig. 43. To draw an involute. Upon the circumference of the given circle mark off any FIG. 42. Drawing a Spiral FIG. 43. Drawing an Invo- about a Regular Polygon. lute. number of equally distant points, as 0-1-2-3, etc., and draw lines tangent to the circle at these points, beginning at point 1. Then with the compasses set the same as for marking off the spaces on the circle, mark off one space on line 1, two spaces on line 2, three spaces on line 3 t and so forth. A curved line starting at and passing through these points will be the required involute. This curve is used for the shape of the teeth of involute gears. Prob. 21, Fig. 44. To draw a cycloid. Upon the base line ab mark off any number of equally dis- tant points, as 0-1-2-3, etc., the distance between 28 SELF-TAUGHT MECHANICAL DRAWING them being made, for convenience sake, about one- sixth of half the circumference of the generating circle. Beginning at 1 erect perpendiculars from these points, and with centers on these lines draw arcs of circles tangent to the base line to represent FIG. 44. Drawing a Cy- cloid. FIG. 45. Drawing an Epicy- cloid. successive positions of the generating circle as it is rolled along. With the compasses set as for spacing off the base line, mark off one space on the arc which starts from point 1, two spaces on arc 2, three spaces on arc 3, and so forth. A curved line starting at and pass- ing through' the points thus obtained will be the required cycloid. An epicycloid, Fig. 45, or a hypocycloid, Fig. 46, is formed in precisely the same way, excepting that as the base line, ab, is an arc of a circle, the center lines from points 1-2-3, etc., are made radial. These three cycloidal curves are used for the shape of the teeth of epicycloidal gears, sometimes called simply cycloidal gears. FIG. 46. Drawing a Hypo- cycloid. GEOMETRICAL PROBLEMS 29 Prob. 22, Fig. 47. To draw a parabola by means of intersecting lines. Draw the axis ax, and on it mark the focus / and the vertex v, and at right angles to it draw the line be at a distance from v equal to the distance of v from/. Across the axis, and at right angles to it, draw a number of lines, 1, 2, 3, 4, 5, 6. Then with radius al, and with center at the focus /, draw arcs intersecting line 1; with radius a2, and with center again on /draw arcs intersecting line 2, and so on. A curved line FIG. 47. Drawing a Parabola. passing through these intersections will be a para- bola. It will be seen from this method of drawing a parabola that any point on it is equally distant from the focus, and from the line be, called the directrix. Prob. 23, Fig. 48. To draw a parabola with a pencil and string. Lay out the axis, the focus, the vertex and the directrix as before. Attach one end of a thread to the focus, / by means of a pin, and attach the other end of the thread to the square shown at d, having the thread of such 30 SELF-TAUGHT MECHANICAL DRAWING length that when the inner edge of the square is on the axis, ax, the thread if drawn down with a pencil will just reach to the vertex, v. Now slide the square along be in the direction of the FIG. 48. Simplified Method of Drawing a Parabola. arrow, keeping the pencil against the square; the thread will cause the pencil to move along so as to describe a parabola as shown. Prob. 24, Fig. 49. To draw a parabola of a given \ FIG. 49. Another Method of Drawing a Parabola. breadth of opening, ab, and of a given depth, cd. Draw ef parallel with ab, and draw ae and bf paral- lel with cd, having ac and be equal. Space off dc GEOMETRICAL PROBLEMS 31 and df into any number of equal parts, and also space off ea and/6 into the same number of equal parts, as shown. From d draw lines to the di- visions on ea and/6, and from 1, 2, 3 and 4 on de and df draw perpendicular lines to intersect the lines drawn from d to 1, 2, 3 and 4 on lines ca and fb. A curved line passing through these inter- sections will be the required parabola. Prob. 25, Fig. 50. To find the focus of a para- bola. Let abed be the given parabola, ef t being its FIG. 50. Finding the Focus of a Parabola. axis. Across the parabola at its vertex, v, draw the line ij at right angles to the axis. From any point, g, on the parabola, draw the line gh parallel to the axis. With center at g.fmd a radius, by trial, which will cut the axis as much inside the vertex, v, as it cuts the line gh beyond the line ij. The intersection at x will be the required focus. CHAPTER IV PROJECTION Mode of Representing Objects. In mechanical drawing, machines, or parts of machines, are rep- resented by views, generally three, in which per- spective is ignored, and which show the object in different positions at right angles to each other. The mode of representing these views, and their positions with regard to one another, which expe- rience has shown to be most convenient is perhaps best shown by means of the familiar cardboard illustration. Let abcdefgh, Fig. 51, represent a piece of cardboard, which we will suppose to be transparent, creased on the dotted lines to permit of the outer portions being turned back. Let us now suppose that we have a prism shaped as shown at C, and of the length shown at A. If the prism is stood upright with its broad side facing the ob- server, and the cardboard, being blank, is held up in front of it, the prism will appear, if all its lines are brought perpendicularly forward to the card- board, as it is shown at A, lines on the prism which would be hidden by its body, as the further corner, being dotted. If section Cof the cardboard is now turned backward through an angle of 90 degrees over the top of the prism we would get the view shown in that part, all lines being brought 32 PROJECTION 33 perpendicularly forward from the prism to the cardboard as before. Likewise if part D of the cardboard were turned backward through an angle of 90 degrees, and the lines of the prism were brought perpendicularly forward onto it, we would get the view shown in that part. The view shown at A is called the elevation, that shown at C is called the plan, and that shown at D is called the side view. Occasionally a piece is so shaped, or FIG. 51. Principle of Projection. has so much of detail to it as to make another side view desirable; such a view would be placed at B. In many other cases, as in the case of the prism here shown, the plan and elevation views alone will fully show the object. The production of these views from one another is called projection ; and by the use of connecting lines, and also at times of temporary construction views, objects may be shown at any desired angle, irregular or curved lines may be traced, and sur- faces may be developed. 34 SELF-TAUGHT MECHANICAL DRAWING An Upright Prism. Fig. 52 shows a prism in its simplest position. A moment's examination will show that the elevation cannot be drawn directly, as the distance apart of the vertical lines which represent the corners of the prism, cannot be deter- mined without other aid; hence it is necessary to draw the plan view first. Horizontal lines having been made to give the height of the prism in the elevation, the vertical lines may then be drawn in FIG. 52. Projections of Prism. FIG. 53. Projections of Tilted Prism. from the plan, as indicated by the vertical dotted line. The Prism Inclined at One Angle. Fig. 53 shows the prism inclined to the right. A brief exami- nation of these views will show that none of them can be drawn directly, as the distance apart of the vertical lines in the elevation and side views is not known, and the lines of the plan view are foreshortened; but the views can be developed from Fig. 52. It is evident that as the prism is tipped, the elevation view will remain unchanged, PROJECTION 35 hence the first step will be to reproduce that view inclined at the desired angle. As the prism is tipped it is also evident that all points in the plan view of Fig. 52 will move in horizontal lines to the right, hence horizontal lines are drawn from these points through the position which the plan will occupy in Fig. 53. The intersection of these lines with vertical lines from the corresponding points in the elevation will determine the position of each point in the plan. The points so determined one by one being then connected by straight lines, gives the plan view as shown. To make the side view, horizontal lines are first drawn from the various points of the prism as seen in the eleva- tion through the position which the side view will occupy. Then, bearing in mind that each point of the prism in the side view will be as much to the left of the vertical line ab as the same point in the plan is below the line ccZ, the position of each point on the horizontal lines is marked off from ab. The Prism Inclined at Two Angles. Fig. 54 shows the prism tipped forward after having been tipped to the right as shown in Fig. 53. An examination of these views will show that not only can they not be drawn directly, but they cannot be devel- oped from Fig. 52. They may, however, be de- veloped from Fig. 53. It is evident that as the prism is tipped forward, the side view of Fig. 53 will remain unchanged ; hence the first step will be to reproduce that view inclined at the desired angle. Next, horizontal lines are drawn from the corners of the prism as seen in this view through 36 SELF-TAUGHT MECHANICAL DRAWING the place which the elevation is to occupy, and the perpendicular line gh is drawn. It is evident that as the prism is tipped forward, the different points of it as seen in the elevation of Fig. 53 do not move any to the right or left, but forward only. Hence, the distance of the corners of the prism from the line ef may be taken by the compasses and marked off from the line gh upon the proper horizontal line. The new position of all of the corners having thus been determined, the con- necting straight lines are drawn, giving the elevation as shown in Fig-, 54. Vertical lines are then drawn from the different points of the prism, as seen in this view, through the posi- tion which the plan is to occupy, and the exact position of each point upon these lines is marked off from mn at the same distance which it is from the line jk in the side view. An Upright Rectangular Prism. The upright rectangular prism shown in Fig. 55 is, of course, drawn in the same way as was the prism shown in Fig. 52. The Prism of Fig. 55 Tipped Forward on One Edge. It is evident that if the prism were to be tipped on its edge in the direction of the arrow No. 1, the result would be the same as though it had been FIG. 54. Projections of Prism Tilted in Two Directions. PROJECTION tipped first to the right, and then directly forward, as was done to produce Fig. 54; but as those angles are not given, the method employed in that case is not readily available. Fig. 56 shows the prism tipped to its new po- sition, and shows, also, the method employed to produce the views. Draw the line cd at the same FIG. 55. Upright Rectan- gular Prism. FIG. 56. Rectangular Prism Tipped Forward. angle to the horizontal as the edge ab of the prism in Fig. 55, and make e/at right angles to it. Upon these lines draw the temporary side view of the prism, A, tipped at the desired angle. With the aid of this view the plan view is readily drawn. Vertical lines are then drawn from the various points of the plan view through the place which the elevation is to occupy, and the exact location of each point is marked off on these lines at the 38 SELF-TAUGHT MECHANICAL DRAWING same height above the base line gh that it is above the line ef in the temporary side view, A. The permanent side view is then developed from the plan and elevation in the same way as was the side view of Fig. 53. Let it now be required to tip the prism of Fig. 55 forward on one corner in the direction of arrow No. 2. It will be seen that tipping it in this direction FIG. 57. Rectangular Prism Tipped in Two Directions. will cause a foreshortening of all of the lines in the plan, hence the use of a single temporary view such as was used in Fig. 56 will not solve the problem; but it may be solved by the use of two temporary views as shown in Fig. 57. Draw the line ab in the direction in which the prism is to be tipped, and the line cd at right angles to it. At A reproduce the plan view of Fig. 55, and at B draw PROJECTION 39 a side view of the prism as it would appear if A were viewed in the direction of the arrow, but in- clined to cd at the required angle. The intersec- tion of lines drawn from the corners of A, parallel with ab, with lines drawn from the same corners of B, parallel with cd, will give their location FIG. 58. Projections of a Cube. in the permanent plan view. This view being finished, the elevation and the permanent side views are drawn in the same way as were those of Fig. 56. Let a cube be set on one corner so that a diagonal of it shall be horizontal; required to show the angle which the edges that meet at that forward corner make with a plane perpendicular to the diagonal, the angle which the sides that have corners coming together at the same point make with the plane, and 40 SELF-TAUGHT MECHANICAL DRAWING also the am ^ ant of foreshortening of the lines which will be caused. In Fig. 58, A shows a face view of the cube set on edge, B shows a side view of the same, and C shows B inclined until the diagonal ke becomes horizontal. The length of ke being laid out on the center line, the position of the other corners is ob- tained as indicated by the arcs a, b, c and d. The angle geh is the required angle which the edges which meet at e make with a plane perpendic- ular to ek, of which fg is an edge view ; the angle fej is the angle which the sides having corners meeting at e make with the plane. D is a face view of C, and any of its lines, when compared with any of the lines of A, will show the fore- shortening caused by the cube being put into this position. The Surface Develop- ment of a Cone. Let A and B, Fig. 59, be the plan and elevation views of a cone. With a radius equal to ab, and with a center at c, draw the arc def, making it equal in length to the circumference of the base of the cone, as shown at A, This may be most conveniently FIG. 59. Development of a Cone. PROJECTION 41 done by spacing it off. Draw the lines cd and cf, and the figure C thus formed will be the required surface development. The Surface Development of a Pyramid Having Its Top Cut Off Obliquely. In Fig. 60, A, B and C show, respectively, the plan, elevation, and side FIG. 60. -Development of a Frustum of a Pyramid. views of the pyramid, the top of which is cut off by the plane ab. These views may be made by the principles already explained, as may also the view at D, which shows the pyramid as though B were viewed in the direction of the connecting dotted line, which is at right angles to ab, thus showing the shape of the section exposed by cutting off the top. 42 SELF-TAUGHT MECHANICAL DRAWING To get the surface development, take a radius equal to the length of one edge of the pyramid as shown at cd in the elevation, this being the only one which shows at full length, the others being more or less foreshortened, and with a center at e in view E, draw an arc of a circle upon which the sides of the base are to be marked off. These points are connected with one another and with e; this gives the shape of the surface of the whole pyramid. Upon the lines connecting the points with e, as el, e2 and e3, the .lengths of the different edges of the cut off pyramid are marked off. As the edge which is seen at the left in the elevation shows full length, its length, dl, may be taken di- rectly and marked off on the line el. As the other edges are seen foreshortened, their lengths cannot be taken directly, but by horizontally transferring the upper end of each edge to the line cd, their actual lengths d2 and d3 may be obtained and then marked off on the lines e2 and e3. The points so obtained being connected, and the outer half sec- tions being finished, gives the required surface development. If the cone shown in Fig. 59 were to have its top cut off obliquely, the views of it corresponding to A, B, Cand D, Fig. 60, and its surface develop- ment, would be obtained by dividing off its base, as seen in the plan, into any number of sides, and then proceeding as though it were a pyramid of that number of sides, until the points correspond- ing to those of Fig. 60 had been located, but then connecting them with curved lines instead of straight lines. PROJECTION 43 Intersecting Cylinders, Fig. 61. Required the line of the intersection, the surface development of the branch, and the shape which the end of the branch would appear to have as seen in the view at the right. First draw the elevation, A, in outline, and as much of the end view, B, as can be directly drawn. FIG. 61. Intersecting Cylinders. Opposite the end of the branch in each of these views, and in line with it, draw circles of the same diameter as the branch, and space off the semi- circumference nearest to it into a number of equal parts, the same number in both cases. From the points so obtained draw lines parallel with the center line of the branch, as shown. From the points where these lines in the view B meet the 44 SELF-TAUGHT MECHANICAL DRAWING circle representing the end of the large cylinder, draw horizontal lines intersecting the lines drawn from C. These intersections will be points through which the line of the intersection of the cylinders is to be drawn. From the points where the lines drawn from C cross the end of the branch, draw horizontal lines intersecting those drawn from D. These intersections will be points through which the line representing the end of the branch is to be drawn. To get the surface development of the branch, first draw the line ab, in E, having it in line with the end of the branch. Make this line equal in length to the circumference of the branch, spacing it off equally each way from the center line OX into the same number of spaces as the semi -circumfer- ence of C was divided into. From these points draw lines parallel with OX, and from the points in the intersection of the two cylinders, previously obtained, draw lines parallel with ab, intersecting these lines. These intersections will be points through which a curved line is to be drawn, thus giving the completed surface development of the branch. In drawing these curved lines through the points of intersection, the irregular curves mentioned in the early part of the chapter on instruments and materials are used. Intersecting Cylinder and Frustum of Cone, Fig. 62. Required line of intersection and surface de- velopment of branch, as before. Draw the elevation, A, in outline, continuing the sides of the conical branch either way until they PROJECTION 45 meet at their vertex, a, on the one hand, and to any convenient points, c and d, on the other. In a similar manner draw as much of the end view, B, FIG. 62. Intersecting Cylinder and Cone. as can be made directly. With centers at a and at 6, and with any convenient radius, draw the arcs c'd and ef, intersecting the extended sides of the conical branch. Then, with centers at the inter- 46 SELF-TAUGHT MECHANICAL DRAWING section of these arcs with the center line of the branch, draw the half ci cries shown, tangent to the extended sides of the branch, and space them off into a number of. equal parts, the same number in each case. From these points draw lines to the vertices a and b. From the points where these lines in the end view, B, intersect the circle repre- senting the end of cylinder, draw horizontal lines to the elevation, A, intersecting the lines drawn from the vertex a to the half -circle cd. The inter- sections will be points through which the line rep- resenting the intersection of- the cylinder and its conical branch is to be drawn. The shape of the end of the branch as seen in the end view, B, is now obtained in the same manner as in the case of the intersecting cylinders. From the points where the lines drawn from the vertex, a, of the side ele- vation A, to the half-circle at cd, cross the end of the branch, draw horizontal lines intersecting the lines drawn from the vertex b. These intersec- tions will give points through which the line rep- resenting the end of the branch in view B is to be drawn. To get the development of the branch as shown at F take a radius equal to the distance from the apex a to the end of the branch as seen in the side elevation, A, and with a center at g draw an arc hi, making the length of the arc equal to the cir- cumference of the end of the branch as shown at E, spacing equally each way from the center line gj, the length and number of the spaces each way being the same as those obtained in spacing off the semicircle at E. Through these points draw lines PROJECTION 47 radiating from g, as shown. On these lines dis- tances are marked off from the arc hi through which the irregular curved line is drawn which gives the development of the branch. The lengths at the middle and at the extremities may, of course, be taken directly from the elevation A, the length kl being the length on the center line, and the length mn being the length at the extremities. The other lengths, being foreshortened, as seen in the elevation A, cannot be taken directly, but are obtained by transferring the points to either kl or mn as shown by the dotted lines, as was done in the case of the pyramid, Fig. 60. To Draw a Helix. A helix is a line of such shape as would be made by winding a thread around a FIG. 63. Drawing a Helix. FIG. 64. The Helix as it Appears in a Screw Thread. cylinder, and having it advance lengthwise on the cylinder at a uniform rate as it is wound around it. In Fig. 63 we have the side and end views of a cylinder upon which it is desired to draw a helix, which shall advance from a to b in making a half turn around it. Divide the space from a to b into any number of equal parts, and at the points so obtained erect perpendicular lines. Divide the 48 SELF-TAUGHT MECHANICAL DRAWING semi-circumference of the end view of the cylinder, toward the side view, into the same number of equal parts, and from these points draw horizontal lines to meet the perpendiculars previously erected. Where these lines meet will be points through which the helix is to be drawn. The outlines of a screw thread are helices. Fig. 64 shows a double threaded Acme standard, or 29 degree threaded screw, the outline of which, on its outside diameter, is the helix of Fig. 63. Isometric Projection. If a cube is tipped over on one corner, so that the diagonal of it is horizontal as shown at D, Fig. 58, and also in Fig. 65, the FIG. 65. Principle of FIG. 66. An Example of Isometric Projection. Isometric Projection. lines of it will all appear of equal length. Draw- ings made on this principle, as Fig. 66, are called isometric drawings. Vertical lines remain ver- tical. Horizontal lines become inclined to the horizontal of the paper at an angle of 30 degrees. Circles appear as ellipses, which may be drawn as shown in the upper square of Fig. 65. From the ends of the "short" diagonals, lines are drawn to the middle of the opposite sides. Where these lines cross the "long" diagonals are located the centers from which the ends of the ellipse may PROJECTION 49 be drawn. The ends of the short diagonals will be centers from which to draw the sides of the ellipse. Irregular curves may be drawn as indicated in Figs. 67 and 68. The figure 2 there shown is first drawn in the desired position in a naturally shaped square, which is then divided off by equally spaced lines into smaller squares. The isometric square is then similarly divided off, and the figure is FIGS. 67 and 68. Method of Transferring Irregular Lines in Isometric Projection. made to pass through the corresponding inter- sections. Isometric drawings differ from perspective draw- ings in that receding lines remain parallel, instead of converging to a vanishing point. They may be measured the same as ordinary drawings in any one of the three directions indicated by the lines of the cube. The foreshortening of the lines caused by tipping the cube into this position is generally ignored. If an isometric drawing is to be shown in connection with ordinary views, however, it should be made on a scale of about 8-10 of an inch to the inch, otherwise it would appear too large. CHAPTER V WORKING DRAWINGS As the object of working drawings is to convey to the workman a clear idea of the appearance and construction of the piece to be made, and as the whole "science" of mechanical drawing has been developed primarily for the purpose of conveying the ideas and thoughts of the designer and drafts- man to the men who carry out these ideas in wood and metal, the subject of working drawings is of supreme importance to all mechanics. A working drawing should be as complete as possible, so com- plete, in fact, that when it has once passed out of the draftsman's- hand into the shop, no further questions will be necessary. In or.der to accom- plish this, all necessary information, of whatever kind, should be .included, and, if required, short notes and directions may be written on the draw- ing to prevent eventual misunderstandings. The number of views necessary to properly rep- resent an object must be left for the draftsman's judgment to determine. Usually two views are sufficient, when the object is simple, but when at all complicated, three or more views will be found necessary. Cylindrical pieces can often be ade- quately represented by a single view, on which the various diametral and length dimensions are given. 50 WORKING DRAWINGS 51 While it is customary to put the plan view of an object above the elevation, it frequently becomes necessary, in order to present the objects shown in as clear a manner as possible, to deviate from this rule. A case of this kind is shown in Fig. 69, where the shaft hanger illustrated has been se- lected as an example of the methods employed in working drawings. An examination of the hanger will show that if the plan were placed above the elevation, and if it were represented according to the methods already explained, the box and the yoke with its adjusting screws and check-nuts would have to be shown mostly by dotted lines. Such a multiplicity of dotted lines would tend to confusion; hence the object in view, that of presenting the hanger in as clear a manner as possible, is best accomplished in a case like this by having the plan underneath the elevation, and letting it be a bottom view instead of a top view. In designing a machine detail of this kind, the starting point would of necessity be the shaft itself, and the first step would be to design the box; next would come the yoke, and lastly, the frame. Much of the preliminary work may fre- quently be done on scrap paper; having determined the size and proper proportions of the various parts, the position which the different views will occupy in the finished drawing is easily ascer- tained. The center lines are then laid out as shown, and the drawing built up about these lines as a base. When a drawing is for temporary use only, and 52 SELF-TAUGHT MECHANICAL DRAWING the mechanism represented on it of a simple nature, the assembly drawing, corresponding to the three views in Fig. 69, will answer all purposes, the di- mensions being given directly on this drawing. In FIG. 69. Shaft Hanger. some cases only the most important dimensions would be given, those of secondary consequence being left for the workman to be obtained by " scaling' ' the drawing. This procedure, however, is possible only when the drawing is made care- fully to scale, and is not one that should be en- WORKING DRAWINGS 53 couraged. In general, a drawing should be so di- mensioned that it can be worked to without the workman obtaining any measurements by "scal- ing" the drawing. In most cases it is not possible to show the de- tails of a mechanism clearly enough in an assembly drawing; for if the device shown is more or less complicated, a hopeless confusion results from the attempt to put in all the lines necessary to fully show all the details ; neither would it be possible, for the same reason, to give more than the princi- CAST IRON, BABBITTED FIG. 70. Example of Working Drawing. pal dimensions. In such cases it is, therefore, cus- tomary, after the assembly drawing has been com- pleted, and the proper sizes and proportions of the various parts of the mechanism thus ascertained, to make a separate drawing of each detail, either on the same sheet of paper, or on separate sheets. This permits the parts of the mechanism to be clearly and completely shown and fully dimen- sioned. Figs. 70 and 71 show two pieces of the hanger in Fig. 69 detailed in this manner. These detail drawings give all the required informa- tion for the making of the pieces, and the assembly 54 SELF-TAUGHT MECHANICAL DRAWING drawing merely shows, in a general way, how the parts are to be assembled when completed. In the case of jig and fixture drawings, it is the practice in a great many large drafting-rooms to show assembled views only, and to put all dimen- CAST IRON, Tap %"- 10 thd. FIG. 71. Example of Working Drawing. sions directly on the assembly drawing; the argu- ment advanced in favor of this practice is that ex- perienced pattern-makers and tool-makers, who are, as a rule, the only mechanics who will work on the making of these tools, will find no difficulty in reading the assembly drawing; besides, it is said, WORKING DRAWINGS 55 as a drawing of this kind is, in most cases, used but once, it would be waste of time to have the draftsman detail the different parts of the tool. While these arguments are undoubtedly true in the case of very simple jigs and fixtures, there can be little doubt that in the case of more complicated ones, the comparatively short time required by the draftsman to make detail drawings will be saved many times over in the shop; for the pattern- maker and tool-maker will not have to spend, in the total, a number of hours puzzling over the draw- ing, and even then being liable to make a mistake. In making drawings, it is always a rule to work from the center lines, when the outline of the piece is such that it has a definite center line. Dimensions in either direction from the center line can be best marked off with the compasses. This insures a symmetrical appearance to the fin- ished drawing, such as might not be secured if the dimensions are set off on either side of the center line from the rule, it always being easy to then introduce small errors which show plainly in the finished work. If the piece is of such shape as to have no center line, some one principal line may be selected, one in each direction in each view, and the remaining points and lines may be located from these lines. The various styles of lines ordinarily used in working drawings are shown in Fig. 72. The regular "full" line A A is used for the outlines of objects, and when drawn rather "fine," for cross- hatching or cross-sectioning. The heavy shade line BB is used to represent lines assumed to sepa- 56 SELF-TAUGHT MECHANICAL DRAWING rate the light surfaces of an object from the dark, as will be explained in the following. The dotted line CC, as has already been explained in the pre- vious chapter, is used to represent lines obscured or hidden from view. The line DD, called a "dash" line, is used by a great many draftsmen for dimension lines. Finally, the line EE, the "dash and dot," or, simply, the "dash-dotted" FIG. 72. Styles of Lines Used on Working Drawings. line, is used in common practice for center lines, to indicate sections, etc. This line is also com- monly used for construction lines, in laying out mechanical movements. The dimension lines may be made either fine full lines or "dash" lines, the dashes being about | inch long. A space is left open for the figures giving the dimension. The witness points or ar- row heads, showing the termination of the dimen- sion, are made free hand. Many draftsmen draw the extension and dimension lines in red ink, the arrow heads, however, still being made black. It is well to avoid, as far as possible, having the WORKING DRAWINGS 57 dimension lines cross each other, as such crossing tends to confusion; the difficulty can usually be avoided by having at least one set of dimensions placed outside or between the views, the larger di- mensions being placed farther from the outline of the object than the shorter ones, to avoid having the extension lines of the latter cross the dimen- sion lines of the former. Dimensions under 24 inches are most conveniently given in inches; larger dimensions are given in feet and inches. The usual practice is to indicate feet and inches on drawings by short marks, " prime' ' marks ('), placed at the right, and a little above the figure, one mark (0 indicating feet, and two marks, " double prime' ' marks ("), indicating inches, so that 5' 7" would read 5 feet 1 inches. Some drafts- men do not consider this method of marking safe enough to eliminate mistakes, and prefer to write dimensions of this kind in the form 5 ft. 7". A method equally satisfactory in preventing possible mistakes is to place a short dash between the figure giving the number of feet and that giving the number of inches, at the same time retaining the "prime" marks; thus, 5' 7". When feet only are given, it is well, for the sake of uniformity and to prevent any misunderstanding, to give the dimension in the form 5' 0". A few examples showing the principles of the usual methods of dimensioning drawings may be of value. In Fig. 73 is shown a simple bushing. The diameter of the hole or bore is given as 2 inches by a dimension line passing through the center of the circles in the end view. It is con- 58 SELF-TAUGHT MECHANICAL DRAWING fusing, however, to have more than one dimension line passing through the same center, and, there- fore, the outside diameters of the bushing have been given on the side view. The lengths of the various steps or shoulders of the bushing are given below the side view, as is also the total length. It will be noticed that the dimensions of the three steps are slightly offset that is, the dimension FIG. 73. Simple Example of Dimensioning a Drawing. lines do not extend in one straight li'ne ; this makes a very clear arrangement. The method of dimensioning holes drilled in a circle is shown in Fig. 74. Outside of the dimen- sion for the holes themselves only the diameter of the circle passing through the centers of the holes is given, together with the number of holes. As the holes, of course, are to be equally spaced, that is* all that is required. When a great many bolt holes or bolts occur around a flange, it is not nec- essary to draw them all in on the working draw- ing; a common method is to show a few, and to WORKING DRAWINGS 59 draw the circle passing through their centers, the pitch circle. The total number of bolts around the flange is, of course, also given. A case of this kind is illustrated in Fig. 75. When a great many holes are drilled in a row, a similar expedient may FIG. 74. Dimensioning Holes Drilled in a Circle. be adopted to avoid showing and dimensioning all the holes; an illustration of this is shown in Fig. 76. In Fig. 77 are shown the common methods of dimensioning screws and bolts. At A is shown a hexagon head bolt, so drawn that three sides of 60 SELF-TAUGHT MECHANICAL DRAWING the head are visible. Hexagon bolt-heads are usually drawn in this manner in all views, irre- spective of the fact that the rules of projection would call for only two sides to be visible in one view. The reason for this is partly that the bolt- FiG. 75. Simplified Method of Dimensioning Holes Drilled in a Circle. head looks better when three sides are visible, and partly that when so drawn there can be no confu- sion whether a hexagon or a square head is meant. If only two sides were shown, as at B, the head, especially if carelessly drawn, might be mistaken for a square bolt-head. As a rule, the dimensions WORKING DRAWINGS 61 of bolt-heads are standard for given diameters of bolts, and no dimensions are required for the head. In some cases, however, the head may be required to fit a given size of wrench, or for some other reason be required to be made different from the standard size ; in such cases dimensions may be given as shown at C, Fig. 77, the dimension "V hex." indicating that the head is one inch K 10-HOLE8-2-CENTER-DISTANCE j ! * ~t- --$--&- -6 j< : 10-HOLES-2-CENTER-DISTANOE >j T FIG. 76. Dimensioning Holes Drilled in a Row. "across flats." In the same way, "f" sq." would indicate that the head should be square, and three- quarters inch "across flats." The length of the bolt should be given as shown in the lower view in Fig. 77. The dimensions should be given "under the head," both the total dimension, and the distance to the beginning of the thread. In general, full circles should be dimensioned by their diameters ; an arc of a circle, again, should be dimensioned by its radius. The center from 62 SELF-TAUGHT MECHANICAL DRAWING which the arc is struck should preferably be indi- cated by a small circle drawn around it. In small dimensions, the arrow points are frequently placed outside of the lines between which the dimension is given, as shown in Fig. 71 in dimensioning the narrow ribs; sometimes, the figures giving the "A"7 ^ JL -a 8T D HEAD < -- --- 2M FIG. 77. Dimensioning Screws and Bolts. dimension are themselves placed outside of the space between the arrow heads, because the space is too small to permit the dimension to be clearly written within it. The principal dimensions should be so given that the workman will not have to add a number of other dimensions to get them. When the dimensioning of a piece naturally divides itself into several measurements, an over-all dimension should always be given for verification. If, how- WORKING DRAWINGS 63 ever, the piece terminates with a round end, as the yoke in Fig. 71, the over-all dimension may properly terminate at the center of curvature of the end, the distance beyond being of entirely secondary importance, and being taken care of by its radius. If a dimension has been given in one view, there is usually no reason for repeating it in the other views; sometimes such repetitions would cause too many dimensions to be given in each view, so that confusion would arise, and in- stead of making the drawing plainer, the repeti- tion of dimensions might cause mistakes which otherwise would have been avoided. Drawings should always be dimensioned the full size of the finished article, regardless of the scale to which the drawing is made. If a drawing is made to any other scale than full size, it is cus- tomary to state on the drawing the scale to which it is made, as " Scale, i inch=l ft." A drawing should be so marked as to tell the workman what surfaces are to be finished ; a fin- ished surface is usually indicated by the letter "f" placed either upon the line representing the surface, or in close proximity to it. While the amount and kind of finish is usually left to the workman to determine, the best modern methods require that the draftsman should indicate on the drawing how closely the various parts are to be machined. A very commendable method is to give dimensions in thousandths of an inch, where accuracy is required, and in common fractions in cases where there is no need of working to thou- sandths. In very highly systematized establish- 64 SELF-TAUGHT MECHANICAL DRAWING ments, the limits of variation between which any measurement is allowed to vary, are given with each dimension, or, at least, with dimensions for diameters which are to fit the holes or bores of other pieces. The determination of the limits of accuracy required calls for good judgment on the part of the draftsman. Limits may be expressed in two ways. For instance, a running fit on a shaft to go into a li inch standard size hole may be marked -0.0005 max. l5 -0.0015min. or it may be expressed 1.4995 max. 1.4985 min. which means that the shaft must not be larger than 1.4995 inch, and not smaller than 1.4985 inch. On drawings, the tap drill size and the depth of tapped holes should always be shown. Surfaces to be ground to size should be marked " grind/' If the surface is to be filed, the words "file finish' ' are substituted for the letter "f." Finishing marks, as a rule, are used on castings and forg- ings only. On work made from bar stock, every surface is nearly always finished, so that here the finishing marks are omitted. When a casting or forging is finished on every surface, it is not nec- essary to show finish marks, but the words "finish all over" may be written in a conspicuous place, so as to readily catch the eye of the workman. If, on work made from bar stock, it is desired that the piece be left rough at any point, the words WORKING DRAWINGS 65 "stock size" may be applied to the figures giving that particular dimension. For instance, on a li-inch cold rolled shaft, turned for journals for a short distance at each end, the central part would be dimensioned "li-inch stock size/' While the practice of indicating finished surfaces by the letter "f" is by far the most frequently met with, it is by no means universal. In some shops the words " polish, " "ream," "finish," etc., are written near the lines representing the sur- faces to be thus treated. Still another method much in use is to draw a red line outside of the line representing each surface to be finished. If a blue-print is made from a tracing thus pre- pared, the red lines will print fainter than the black ones, and the finish lines on the blue-prints are traced over with a red pencil or red ink before being sent out in the shop. This method, how- ever, is more expensive than that of indicating the finished surfaces by the letter "f," and on complicated drawings, the many additional red lines tend to cause confusion. By whatever method the finish is indicated, the finishing marks should always be shown fully in every view of the object. It frequently happens that the representation of an object is made clearer by the use of sectional views, representing the object as having been cut in two, either wholly or in part. Examples of this are shown in Figs. 69, 70 and 71. From these illustrations it is apparent that the construction of the various pieces is much more clearly exhibited when a section is shown. The surface "cut" or 66 SELF-TAUGHT MECHANICAL DRAWING shown in section is cross-hatched or cross-sectioned with fine lines at a distance apart varying from a thirty-second to an eighth of an inch, according to the size of the drawing and the piece. The cross-sectioning brings the parts in section into bold contrast with the remainder of the drawing, and prevent all confusion as to what parts are in section and what parts shown in full. All lines beyond the sectional surface which are exposed to view, should be shown in the drawing as usual. Should it be deemed necessary, which it seldom is, to show any parts that have been cut away for the purpose of showing a section, such parts may be drawn in by dash-dotted lines, this indicating that the parts thus shown are in front of the section and actually cut away. When a mechanism is shown in section, the dif- ferent parts of the same pieces should always be cross-sectioned by lines inclined in the same direc- tion, while separate pieces adjoining each other should always, when possible, be cross-sectioned by lines running in different directions. When a solid round piece is exposed to view by a section, it is customary to show, it solid, and not to section it; the screw stud in Fig. 69 is an example of this practice. Sectional views may also be used for many pur- poses where a slight deviation from the theory of projection will tend to simplify the representation of certain machine details. The shape of the arm of a pulley or gear, or of any other part of a cast- ing, may be conveniently represented in this way. The cutting plane may be assumed to lie at any WORKING DRAWINGS 67 angle necessary to bring out the details most clear- ly. A sectional view, for instance, may represent a casting as though it were cut through partly on one plane and partly on another. In all such cases, however, it should be indicated in another view of the object just where the sectional views are sup- SECTION AT G-H FIG. 78. Methods of Showing Sections. posed to be taken, so that no confusion may arise on this account. The examples in the following will serve to make clear the principles laid down. In Fig. 78 are shown sections of two hand- wheels. When an object is symmetrical it is unnecessary to show more than one half in sec- tion, although it is quite common to section gears, pulleys, etc., completely on working drawings. The hand-wheel at A in Fig. 78 is represented as 68 SELF-TAUGHT MECHANICAL DRAWING though cut in two along its diameter BC. When the section is taken along the center line, it is not absolutely necessary to explain where the section is taken ; but it can do no harm to make a practice of in all cases to state where the section is made, except when perfectly obvious. In this case it would be clear that the section is taken through SECTION AT A-B FIG. 79. A Gear-wheel in Section. the center, and the legend "Section at BC' 9 is given only to show the principle. The hand-wheel at D is provided with four arms, and the method of representing the shape of the arms, hub and rim are clearly indicated. In Fig. 79 are shown two views of a gear-wheel, indicating the conventional method of represent- ing gears on drawings. The view on the left side is the side view, and as all the teeth are, of course, WORKING DRAWINGS 69 alike, it is unnecessary to draw more than a few of them. The pitch line of the teeth is represented by a dash-dotted line. In the part of the gear- wheel rim where the teeth are not shown, the face of the gear is indicated by a solid line, and the bottom of the teeth by a dotted line. In the case of machine-cut gearing, where the teeth are cut by standard formed cutters, it is unnecessary to show any teeth at all on the rim of the gear, it being sufficient to state the pitch and the number of teeth, as will be more fully explained later in the chapter on gearing. To show the shape to which the arms are formed, a sectional view of one of the arms is drawn in the side view; the ends of the shaft are supposed to be broken off, and are, therefore, sectioned as shown. The right- hand view of the gear is a section taken along the line AB. It will be noted that the shaft and key are not sectioned, usual practice being followed in this respect. The gear shown has five arms, and the line AB cuts through one of them only. This arm, however, is not sectioned in the right-hand view, and two opposite arms are drawn as though both of them lay in the plane of the paper. While this is not theoretically correct, it is the method usually followed because of simplicity in drawing and clearness of representation. The method of representing the gear teeth in the sectional view is the one commonly employed. Sectional and top views of a cylinder end with flange and cover are shown in Fig. 80. This cylinder cover has only five bolts, and the plane through which the section is taken cuts through 70 SELF-TAUGHT MECHANICAL DRAWING only one of the bolts. It is common practice, how- ever, to draw the section as shown at the left. The bolts are shown as if two of them were in the plane of the section. The bolts are not sectioned, FIG. 80. Section of Cylinder End with Flange and Cover. but are drawn in full, as explained previously. Dotted lines of the remaining bolts, or full lines of their nuts, should not be shown, because this detracts from the clearness of the drawing; the top view shows clearly the number of the bolts and their arrangement, and that is all that is nec- essary. Some draftsmen prefer to draw sections WORKING DRAWINGS 71 of this kind as indicated at the right in Fig. 80. This method, however, is not as commonly used. In a case where the object is rather unsymmet- rical, as, for instance, in Fig. 81, the draftsman's judgment must often be relied upon to decide how |< A -< A FlG. 81. Another Method of Showing Sections. it shall best be shown in section. Usually the sectional view is made symmetrical as shown, the distances A in the lower view being made equal to the radius A in the top view, The materials for the various details making up a complete mechanism are usually cross-sectioned CF "HE UNIVERSITY OF 72 SELF-TAUGHT MECHANICAL DRAWING in such a way as to indicate the material from which each piece is made. There is, however, no universally adopted or recognized standard for cross-sectioning for the purpose of indicating dif- ferent materials. In Fig. 82 is shown a chart, FIG. 82. Cross-sectioning used for Indicating Different Materials. published by Mr. I. G. Bayley in Machinery, Oc- tober, 1906, which represents average practice, although it must be distinctly understood that there is no agreement in all respects between the numerable charts in use in various drafting-rooms. For this reason, cross-sectioning alone should never be depended upon for indicating to the work- WORKING DRAWINGS 73 man the kind of material to be used. Written directions should also be given, the kind of mate- rial for each part being plainly marked. Tool steel may be abbreviated "T. S.", machine steel, "M. ROUND BAR, SOLID ROUND BAR, HOLLOW SQUARE OR RECTANGULAR BAR WOODEN BEAM FIG. 83. "Broken' I-BEAM Drawings of Long Objects. S."; wrought iron, "W. L"; cast iron, "C. L", etc. The less common materials in machine con- struction, such as bronze, brass, copper, etc. , should preferably be written out in full, in order to avoid any chances for confusion. It is better to be too 74 SELF-TAUGHT MECHANICAL DRAWING explicit as regards the information on the draw- ing, than to risk misunderstandings and conse- quent errors. Long bars, shafting, structural beams, etc. , can- not conveniently be shown for their full length on the drawing. In such cases the pieces are drawn as long as the drawing and the adopted scale per- mit, and are broken as shown in Fig. 83, a part between the two end portions shown being imag- ined as broken out. The di- mensions, of course, are given for the full length of the piece, as if not broken. There are several conven- tional methods for showing screw threads; these methods are adopted largely for saving of time, as it would be out of the question to spend the time required for drawing a true helical screw thread on a work- ing drawing. A method for very nearly approximating the appearance of a theoretically correct screw drawing is shown in Fig. 84, where the projection of the screw helix is drawn by straight lines. The V- shaped outline is first laid out, and the connecting lines are then drawn. It will be noticed that the lines representing the roots of the threads are not parallel with those representing the tops or points. This aids in making the drawing resemble that of a true helix. Usually, however, much simpler methods are FIG. 84. Method of Drawing a Screw, Giving Correct He- lix Effect. WORKING DRAWINGS 75 employed for indicating screw threads. In Fig. 85, A, B and C, some of these methods are shown. When a long piece is threaded the entire length, this fact can be indicated as at D, which saves drawing the conventional thread for the full length of the piece. The lines indicating the thread are L.H E F FIG. 85. -Simplified Methods for Showing Screw Threads. inclined, the same as would be the lines represent- ing the true helix. At E in Fig. 85 is shown a right-hand thread and at F a left-hand thread, the different direction of inclination of the thread in- dicating this fact. However, if a thread is to be left-hand, it should always be so marked on the drawing. It is usual to abbreviate left-hand, writ- ing "L. H." 76 SELF-TAUGHT MECHANICAL DRAWING Three methods of indicating tapped holes are shown in Fig. 86, these being used when the holes are obscured from view, and shown by dotted lines. When a tapped hole is shown in section, and looked upon from the top, it is shown as indicated at D, while if seen from the side, in section, it is repre- FIG. 86. Simplified Methods for Indicating Tapped Holes. sented as at E. A surface having tapped holes in it, seen from above, is shown at F. At G and H are shown the methods of representing bolts or screws inserted in place in tapped holes. It will be noted that when the threads of a tapped hole are exposed to view by section, the lines repre- senting the screw helix will be seen to slope in the opposite direction to those of the screw, it being WORKING DRAWINGS 77 the back side that is exposed to view. An example of this is shown in Fig. 71 as well as in Fig. 86. In drawings made for use in the shop it is cus- tomary to make the lines of uniform thickness. For shop use such drawings are as good as any. When, however, the purpose of a drawing is chiefly to show up the object which it represents, its ef- fectiveness may be considerably enhanced by the use of shade lines as shown in Fig. 87. In shade line work, the light is usually assumed to come from the upper left hand corner, and to shine diagonally across the paper at an angle of forty- five degrees. Lines on the side of the object away from the light, or lines separating light from dark surfaces, are made extra heavy. This gives to the FlG 87> _ Use O f drawing a suggestion of relief. shade Lines. An examination of the lines of Fig. 87 taken in connection with the direction from which the light is supposed to come will show, without the aid of any other view, that the hex- agonal part is raised above the surface of the square, and that the circle in the center represents a depression. When a drawing is intended for permanent use it is customary to make only a pencil layout on paper, usually on brown paper, and from this to make a tracing from which any number of blue print copies may be made. The tracing is usually made on the regular tracing cloth. This has one glazed and one unglazed surface. Either surface 78 SELF-TAUGHT MECHANICAL DRAWING may be used. The tracing cloth is drawn tightly over the pencil drawing, and its surface is cleaned of any greasiness with dry powdered chalk. This insures a good flow to the ink. In doing the ink work curved lines should be made first, straight lines afterwards, as mentioned in Chapter I. The blue prints are made in the same manner as photographs are printed, the tracing taking the place of the photographic negative. An exposure of from three to ten minutes may be required, de- pending on the freshness of the blue print paper and the brightness of the sun. After the proper exposure has been given, which may require some experimenting at first, until one gets accustomed to the change in the paper which the light makes, the print is thoroughly rinsed out in clear water and dried, by being hung up by one edge. White writing may be made on a blue print with saleratus water, the water being given all the sale- ratus it will dissolve. CHAPTER VI ALGEBRAIC FORMULAS IN order to be able to carry out the calculations required in simple machine design, it is necessary that a general understanding of the use of for- mulas, such as are used in mechanical hand-books and in articles in the technical press, is acquired. Knowledge of algebra or so-called " higher mathe- matics" is by no means necessary, although, of course, such knowledge is very valuable ; but simple formulas can be used, and the results of scientific results employed in practical work to a very great extent, by any man who understands how to use the formulas given by the various authorities ; and the knowledge required for an intelligent use of algebraic formulas can be very easily acquired. All the mathematical knowledge necessary as a foundation is a clear understanding of the funda- mental rules and processes of arithmetic. A formula is simply a rule expressed in the sim- plest and most compact manner possible. By using letters and signs in the formula instead of the words in the rule, it is possible to condense, in a very small space, the essentials of long and cum- bersome rules. The letters used in formulas sim- ply stand in place of the figures which would be used for solving any specific problem ; the signs used are the ordinary arithmetical signs used in 79 80 SELF-TAUGHT MECHANICAL DRAWING all kinds of calculations. As each letter stands for a certain number or quantity, whenever a specific problem is solved the figures for that case are put into the formula in place of the letters, and the calculation is carried out as in ordinary arithmetic. This may, perhaps, be made clearer by means of a few examples. The circumference of a circle equals the diameter times 3.1416. This rule may be written as a formula as follows : C= DX 3.1416. In this formula C = circumference, and D = diameter. No matter what the diameter is, this formula says, the circumference is always equal to the diameter (D) times 3.1416. Assume that the diameter is 5 inches. Then, to find the circumfer- ence, place 5 in the formula in place of D. C = 5 X 3.1416 = 15.708 inches. If the diameter of a circle is 12 feet, then C = 12 X 3.1416=37.6992 feet. This, of course, is the very simplest kind of a formula, but it illustrates the principle involved, and indicates how easily formulas may be em- ployed. One of the most well-known formulas in steam engineering is that giving the horse-power of an engine, when the average or mean effective pres- sure of the steam on the piston, the length of the stroke of the piston in feet, the area of the piston in square inches, and the number of strokes per minute, are known. Let ALGEBRAIC FORMULAS 81 H.P. = horse-power, P = mean effective pressure in pounds per square inch, L = length of stroke in feet, A = area of piston in square inches, and N = number of strokes per minute. Then PX LX A X N H.P. 33,000 The rule conveying this information expressed in words would require considerable space, and be difficult to grasp immediately ; but the meaning of the formula is quickly understood. If the pressure (P) equals 75 pounds, the stroke (L) 2 feet, the area of the piston (A) 125 square inches, and the number of strokes per minute (N) 60, then TT D 75 X 2 X 125 X 60 OA H ' P ' = ~~ It will be seen that the values for the different quantities are merely inserted in the formula in place of the corresponding letters, and then the calculation is carried out as usual. It will be remembered that the line between numerator and denominator in a fraction also means a division; that is i i OK f)(\f) ^ = 1,125,000 -* 33,000 - 34.1. It is very common in formulas to leave out, en- tirely, the sign of multiplication ( X ) between the letters expressing the values of the various quanti- ties that are to be multiplied. Thus, for example, 82 SELF-TAUGHT MECHANICAL DRAWING PL means simply P X L, and if P = 21 and L = 3, then PL = P X L = 21 X3 = 63. If the multipli- cation signs are left out in the formula for the horse-power of engines just referred to, the for- mula PXLXA XN ,, , ' ... PLAN - could be written As a further example of the leaving out of the multiplication sign in a formula, assume that D = 12, R = 3, and r = 2, then DRr DXRXr = 12 X 3 X 2 72 _ 9 9 9 : 9 : It must be remembered that no other signs, ex- cept the multiplication sign, may thus be left out between the letters in a formula. From the examples given, the use of simple formulas is clear; each letter stands for a cer- tain number or quantity which must be known in order to solve the problem ; when the formula is used for the solution of a problem, the letters are simply replaced by the corresponding number, and the result is found by regular arithmetical operations. The expressions "square" and "square root" and "cube" and "cube root" are frequently used in engineering hand-books and technical journals. It would seem, to one unfamiliar with these names and their mathematical meaning, as well as the signs by which they are indicated, that difficult mathematical operations are involved; but this is not necessarily always the case. The square of a number is simply the product of that number mul- ALGEBRAIC FORMULAS 83 tiplied by itself. Thus the square of 3 is 3 X 3 = 9, and the square of 5 is 5 X 5 = 25. In the same way, the square of 81 is 81 X 81 = 6561. Instead of writing 81 X 81, it is common practice in mathematics to write 81 2 , which is read "81 square/' and indicates that 81 is to be multiplied by itself. Similarly, we may write 7 2 = 7 X 7 = 49, and 12 2 = 12 X 12 = 144. The little "2" in the upper right-hand corner of these expressions is called "exponent." Nearly all mechanical and engineering hand-books are provided with tables which give the squares (and also the square root, cube and cube root) of all numbers up to 1000, so that it is usually unnecessary to calculate these values by actual multiplication. As the squares of numbers are frequently used in formulas for solving problems occurring in machine design and machine-shop calculations, a few examples will be given below of formulas con- taining squares. The area of a circle equals the square of the radius multiplied by 3.1416. Expressed as a for- mula, if A = area of circle, R = radius, and the Greek letter n (Pi) = 3.1416, we have: A = E 2 K. If we want to know the area of a circle having a 5-foot radius, we have : A = 5 2 7r=5X5X 3.1416 - 78.54 square feet. As a further example, assume a formula to be given as follows : A _ D 2 N + R 2 n A ~ DR 84 SELF-TAUGHT MECHANICAL DRAWING Assume that D = 3, N = 5, R = 4, and n (as usual) =3.1416. What is the value of A? Insert- ing the values of the various letters in the formula, we have : 3 2 X 5 + 4 2 X TT 3X3X5 + 4X4X7T A = 3X4 3X4 9 X 5 + 16 X n _ 45 + 50.2656 _ 95J2656 _ _ QQQQ 12 12 12 It will be seen in the example above that all the multiplications are carried out before any addition is made. This is in accordance with the rules of mathematics. When several numbers or expres- sions are connected with signs indicating that additions, subtractions, multiplications or divisions are to be made, the multiplications should be carried out before any of the other operations, because the numbers that are connected by the multiplication sign are actually only factors of the product thus indicated, and consequently this product must be considered as one number by itself. The other operations are carried out in the order written, except that divisions when written in line with additions and subtractions, precede these operations. A number of examples of these rules are given below: 12 X 3 + 7 X 2i - li= 36 + 17J - 1J = 52. 5 + 13X7-2=5 + 91-2 = 94. * 9-3 + 9X3=3 + 27 = 30. 9 + 9-3-2=9 + 3-2 = 10. Sometimes, however, in formulas, it is desired that certain operations in addition and subtraction ALGEBRAIC FORMULAS 85 precede the multiplications. In such cases use are made of the parenthesis ( ) and bracket [ ]. These mathematical auxiliaries indicate that the expres- sion inside of the parenthesis or bracket should be considered as one single expression or value, and that, therefore, the calculation inside the parenthe- sis or bracket should be carried out by itself com- plete before the remaining calculations are com- menced. If one bracket is placed inside of another, the one inside is first calculated, and when com- pleted the other one is carried out. Some examples will illustrate these rules and principles: (6 - 2) X 3 + 4 =4 X 3 + 4 = 12+ 4 = 16. 3 X (12 + 7) - 28i = 3 X 19 - 28i = 57-28J =2. 3 + [5 X 3 (5 + 2) - 3] X 6 = 3 + [5 X 3 X 7 -3] X 6 = 3 + [105 - 3] X 6 = 3 + 102 X 6 = 3 + 612 = 615. Without the parentheses and brackets, the calcu- lations above would have been as follows : 6-2X3 + 4 = 6-6 + 4 = 4. 3 X 12 + 7 H- 28i = 36 + 0.2456 = 36.2456. 3 + 5X3X5 + 2-3X6 = 3 + 75 + 2- 18 = 62. These examples should be carefully studied until thoroughly understood. We are now ready to return to the question of square roots. The square root of a number is that number which, if multipled by itself, would give the given number. Thus, the square root of 9 is 3, because 3 multiplied by itself equals 9. The square root of 16 equals 4, of 36 equals 6, and so forth. It will be seen at once that the square root may be 86 SELF-TAUGHT MECHANICAL DRAWING considered, or, rather, actually is the reverse of the square, so that if the square of 20 is 400, then the square root of 400 is 20. In the same way, as the square of 100 is 10,000, so the square root of 10,000 is .100. The sign used_m mathematical formulas for the square root is V . Thus V 9 = 3, V 49 = 7, and . so forth. The process of actually calculating the square root is rather cumbersome, and it is very seldom required, because, as already mentioned, the engineering hand-books usually give tables of square roots for all numbers up to 1000, and for larger numbers the tables can also be used for obtaining the square root approximately correct, or at least near enough so for almost all practical calculations. The cube of a number is the product resulting from repeating the given number as a factor three times. Thus, the cube of 3 is 3 X 3 X 3 - 27, and the cube of 17 is 17 X 17 X 17 = 4913. In the same way as we write 2 2 = 2X2 = 4, for the square of 2, so we can write 2 3 = 2 X2 X 2 = 8, for the cube of 2. The exponent ( 3 ) indicates how many times the given number is to be repeated as a factor. The cube of 4, for example, may be written 4 3 = 4 X 4 X 4 = 64. Similarly 17 3 - 4913. The expres- sion 17 3 may be read " the cube of 17," "17 cube/' or "the third power of 17." In the same way as the square root means the reverse of square, so the cube root (or "third root") means the reverse of cube or "third power" ; that is, the cube root of a number is the number which, if repeated as factor three times, would give the given number. For example, the cube root of 64 is 4, because 4 X 4 X ALGEBRAIC FORMULAS 87 4 = 64. It is evident that if the cube of a number, say 6, is 216 (6 X 6 X 6 = 216), then the cube root of 216 is 6. The sign used injformulas for the cube root is f^. For_example, f 7 8 = 2 (because 2X2 X 2= 8), and 1^125 = 5 (because 5X5X5= 125). Similarly, 1^3,723,875 = 155. The use of the square and square root, and cube and cube root in formulas may be shown by a few examples : V B X V C Assume that B = 27, C = 25, and D = 2. Insert these values in the formula. Then X 125 3X5 15 25 2 + 2 2 " 125 +4 "" 129 " As another example: z? A B 3 X V C Assume B = 2, C = 9, and D = 4. Then _ 2 2 + 9 2 +4 2 _ 4 + 81 + 16 _. 101. 2 3 XT/y 8X3 24 In the same way as 2 2 = 2 X 2 = 4, so 2 4 = 2 X 2 X 2 X 2 = 16, and 2 5 = 2X2X2X2X2 = 32. The expression 2 4 is read the "fourth power of 2," and 2 5 the "fifth power of 2." The exponents ( 4 ) and ( 5 ) indicate how many times the given number is to be repeated as factor. If, again, it is required to find the number which, if repeated as factor four times, gives the given number, we must obtain the "fourth root" or V~ 88 SELF-TAUGHT MECHANICAL DRAWING Thus, Vl6 = 2, ^because 2 X 2 X 2 X 2 = 16. In the same way V256 = 4. The fifth root is writ- ten V ; and \/243 = 3, because 3X3X 3X3X3 = 243. These explanations, when fully understood, will eliminate all difficulties with formulas of a simple nature, and with such expressions as cube root, exponents, etc. An important method facilitating the use of formulas, is commonly known as the transposition of formulas. A formula for finding the horse- power which can safely be transmitted by a gear of a given size, running at a given speed, is : D X NX PX FX20Q H.P. = 126,050 In this formula H.P. = horse-power, D = pitch diameter, N = revolutions per minute, P = circular pitch of gear, F = width of face of gear. Assume, for example, that the pitch diameter of a gear is 31.5 inches, the number of revolutions per minute 200, the circular pitch 1J inch, and the width of the face 3 inches. Then, if these values arc inserted in the formula, we have : 31.5 X 200 X lj X 3 X 200 , c H ' R - ~ = 45 power, very nearly. Assume, however, that the horse-power required to be transmitted is known, and that the pitch of the gear is required to be found. Assume that ALGEBRAIC FORMULAS 89 tf.P = 30; Z> = 31.5; N = 200; F=3; and that P is the unknown quantity; then, inserting the known values in the formula, gives us: 31.5 X 200 XPX 3 X 200 126,050 In order to be able to find P, we want it given on one side of the equals sign, with all the known quantities on the other side. If we multiply the expressions on both sides of the equals sign by the same number we do not change the conditions ; thus on v i oa AKA - 31.5 X 200 X P X 3 X 200 X 126,050 1Zb ' Ut 126,050 By canceling the number 126,050 on the right- hand side we have : 30 X 126,050 = 31.5 X 200 X P X 3 X 200. If we now divide on both sides of the equals sign with 31.5 X 200 X 3 X 200, we have: 30 X 126,050 = 31.5X200 XPX 3X200 31.5 X 200 X 3 X 200 31.5 X 200 X 3 X 200 We can now cancel all numerical values in the fraction on the right-hand side; then: 30 X 126,050 p 31.5X200X3X200 This is then the transposed formula giving P, and from this we find that P = 1 inch. In general, any formula of the form B A---C. can be transposed as below : A XC = B C = - 90 SELF-TAUGHT MECHANICAL DRAWING It will be seen that the quantities which are in the denominator on one side of the equals sign, are transposed into the numerator on the other side, and vice versa. Examples: BX C A D . Then: n _ Bx c D AX D n AX D A ' = C ' = ~~B ' A _ EXFX G KXL Then: E _A XKXL F _A XKXL . r _A X KxL FX G ' EX G ' EXF ' rf = EX FX G T EXFX G AXL AXK The principles of transposition of formulas can best be grasped by a careful study of the examples given. Note that the method is only directly ap- plicable when all the quantities in the numerator and denominator are factors of a product. If con- nected by + or - signs, the transposition cannot be made by the simple methods shown unless the whole sum or difference is transposed. Example : A =- ; then D =~and + C= A X D. The most usual caclulations, perhaps, in some classes of machine design, are those involving the finding of the strength of certain machine mem- bers ; and, in order to find the strength qf these ALGEBRAIC FORMULAS 91 members, it is necessary to first find the cross- sectional area of the part subjected to stress. For this reason, the remainder of this chapter will be largely taken up with rules and formulas for find- ing the areas and other properties of various geo- metrical figures. Rules and formulas for volumes of solids will also be given. Examples have been given in some cases merely to show the applica- tions of the formulas. The area of a triangle equals one-half the prod- uct of its base and its altitude. The base may be any side of the triangle, and the altitude is the length of the line drawn from the angle opposite the base, perpendicular to it. Assume that A = area of triangle, . B = base, - H = altitude. Then the rule above may be expressed as a formula as follows : A ^BXH ' Let the base (B) of a triangle be 5 feet, and the altitude (H) 8 feet. Then the area 5X8 40 ' . A = n = -JT = 20 square feet. z z The area of a square equals the square of its side. If A = the area, and S the side of the square, then If the side is 9.7 inches long, then A = 9.7 2 = 9.7 X 9.7 =94.09 square inches. 92 SELF-TAUGHT MECHANICAL DRAWING The area of a rectangle equals the product of its long and short sides. If A = area, L = length of the longer side, and H= length of the shorter side, then A = L X H. The area of a parallelogram equals the product of the base and the altitude. The area of a trapezoid equals one-half the sum of the parallel sides multiplied by the altitude. If A = area, B = length of one of the parallel sides, C = length of the other parallel side, and H = altitude, then B Assume that the lengths of the two parallel sides are 12 and 9 feet, respectively, and that the altitude is 16 feet. Then A = ^-^ X 16 = 10.5 X 16 = 168 square feet. To find the area of an irregular figure bounded by straight lines, divide the figure into triangles, and find the area of each triangle separately. The sum of the areas of all the triangles equals the area of the figure. The circumference of a circle equals its diameter multiplied by 3.1416. The diameter of a circle equals the circumfer- ence divided by 3.1416. The area of a circle equals the square of the diameter multiplied by 0.7854. The diameter of a circle equals the area divided ALGEBRAIC FORMULAS 93 by 0.7854, and the square root extracted of the quotient. If D = diameter, C = circumference, and A = area, these last rules may be expressed in formulas as follows : C-DX 3.1416. =3ib- A = D*X 0.7854. D = The length of a circular arc equals the circum- ference of the circle, multiplied by the number of degrees in the arc, divided by 360. If L = length of arc, C = circumference of circle, and N = num- ber of degrees in the arc, then r CXN 360 The area of a circular sector equals the area of the whole circle multiplied by the quotient of the number of degrees in the arc of the sector divided by 360. If a = area of sector, A = area of circle, and N = number of degrees in sector, then a = A X N 360* The area of a circular segment equals the area of the circular sector formed by drawing radii from the center of the circle to the extremities of the arc of the segment, minus the area of the triangle formed by these radii and the chord of the arc of the segment. The area of a pentagon (regular polygon having 94 SELF-TAUGHT MECHANICAL DRAWING five sides) equals the square of the side times 1.720. The area of a hexagon (regular polygon having six sides) equals the square of the side times 2.598. The area of a heptagon (regular polygon having seven sides) equals the square of the side times 3.634. The area of an octagon (regular polygon having eight sides) equals the square of the side times 4.828. The volume of a cube equals the cube of the length of its side. The volume of a prism equals the area of the base multiplied by the altitude. The volume of a cylinder equals the area of its base circle multiplied by the altitude. The volume of a pyramid or cone equals the area of the base times one-third the altitude. The area of the surface of a sphere equals the square of the diameter multiplied by 3.1416. The volume of a sphere equals the cube of the diameter times 0.5236. The volume of a spherical sector equals two- thirds of the square of the radius of the sphere multiplied by the height of the contained spherical segment, multiplied by 3.1416. If V = volume of sector, R = radius of sphere, and H= height of the contained spherical segment, then V = f-tf 2 X HX 3.1416. o Assume that the length of the radius of a spheri- ALGEBRAIC FORMULAS 95 cal sector is 6 inches, and the height of the con- tained segment 2 inches. Then V = -f-X 6 2 X 2 X 3.1416 = 150. 7968 cubic inches. o The volume of a spherical segment equals the radius of the sphere less one-third the height of the segment, multiplied by the square of the height of the segment, multiplied by 3.1416. If R = radius, H = height, and V = volume of segment, then (R- y) XH 2 X 3.1416. Assume that the length of the radius is 4 inches, and the height of the segment 3 inches. Then V = (4 - y) X 3 2 X 3.1416 = 84. 8232 cubic inches. The area of an ellipse equals the long axis multi- plied by the short axis, multiplied by 0.7854. If the area =A, the long axis =B, and the short axis = C, then A = BXCX 0.7854. If the long axis is 12 inches and the short axis 8J inches, then A = 12 X 8i X 0.7854 - 78.54. Formulas and application of formulas have not been given for such rules which are so simple and easy to understand that the reader without diffi- culty can formulate his own formula. CHAPTER VII ELEMENTS OF TRIGONOMETRY TRIGONOMETRY is a very important part of the science of mathematics, and deals with the deter- mination of angles and the solution of triangles. In order to fully understand the subjects treated of in the following, it is necessary that the reader is fully familiar with the usual methods of desig- nating the measurements or sizes of angles. While mathematicians employ also another method, in mechanics angles are measured in degrees and subdivisions of a degree, called minutes. The minute is again subdivided into seconds, but these latter subdivisions are so small as to permit of being disregarded in general practical machine design. A degree is 1-360 part of a circle, or, in other words, if the circumference of a circle is divided into 360 parts, then each part is called one degree. If two lines are drawn from the center of the circle to the ends of the small circular arc which is 1-360 part of the circumference, then the angle between these two lines is a 1-degree angle. A quarter of a circle or a 90-degree angle is called a right angle. The meaning of obtuse and acute angles has already been explained in Chapter II. Any angle which is not a right angle is called an oblique angle. 96 ELEMENTS OF TRIGONOMETRY 97 A minute is 1-60 part of a degree, and a second 1-60 part of a minute. In other words, one circle = 360 degrees, one degree = 60 minutes, and one minute = 60 seconds. The sign () is used for in- dicating degrees; the sign (') indicates minutes, and the sign ( " ) seconds. A common abbreviation for degree is ' l deg. ' ' ; for minute, ' ' min. ' ' ; and for second, "sec." Two angles are equal when the number of de- grees they contain is the same. If two angles are both 30 degrees, they are equal, no matter how long the sides of the one may be in relation to the other. Of all triangles, the right-angled triangle occurs most frequently in machine design. A right-ang- led triangle is one having the angle between two sides a right angle; the angles between the other sides may be of any size. In the calculations in- volved in solving right-angled triangles, a useful application of the squares and square roots of numbers is also presented. Assume that the lengths of the sides of a right-angled triangle, as shown in Fig. 88, are 5 inches, 4 inches, and 3 inches, respectively. Then 5 2 = 4 2 + 3 2 , or 25 = 16 4- 9. This relationship between the three sides in a right-angled triangle holds good for all right-ang- led triangles. The square of the side opposite the right angle equals the sum of the squares of the sides including the right angle. Assume, for ex- ample, that the lengths of the two sides including the right angle in a right-angled triangle are 12 98 SELF-TAUGHT MECHANICAL DRAWING and 9 inches long, respectively, as shown in Fig. 89, and that the side opposite the right angle, the hypotenuse, is to be found. We then first square the two given sides, and from our rule, just given, we have that the sum of the squares equals the square of the side to be found. The square root -* H h ' of the sum must then equal the side, itself. Carry- ing out this calculation we have: 12 M-_9 2 = 144 + 81 = 225 V 225 = 15 inches = length of hypotenuse. Similar methods may be employed for finding any of the sides in a right-angled triangle if two sides are given. If the hypotenuse were known to be 15 inches, and one of the sides including the right angle 9 inches, as shown at D in Fig. 90, then the other side including the right angle can be found. In this case, however, we must subtract the square of the known side including the right ELEMENTS OF TRIGONOMETRY 99 angle from the square of the hypotenuse to obtain the square of the remaining including side. We, therefore, have: 15 2 -9 2 = 225-81 = \/144 = 12 inches = length of unknown side. In the same way, if the lengths 15 and 12 were k; SIDE TO BE FOUND > FIG. 90. known, we could find the side AC, as shown at E, Fig. 90: 15^- 12 2 = 225- 144 = 81 Vgf = 9 inches = length of AC. From these examples we may formulate rules and general formulas for the solution of right- angled triangles when two sides are known. In Fig. 91, at F, the square of AB plus the square of AC equals the square of EC; the square of EC minus the square of AC equals the square of AB; and the square of EC minus the square of AB 100 SELF-TAUGHT MECHANICAL DRAWING equals the square of AC. These rules written as general formulas would take the form : BC 2 -AB 2 = From these formulas we have, by extracting the square root on each side of the equal sign : EC = V~AB 2 + AC 2 AB = VBC 2 - AC 2 AC = V BC 2 - AB 2 These formulas make it possible to find the third side when two sides are given, no matter what the numerical values of the length of the sides may be. Assume AB = 12, and BC = 20; find AC. Ac- cording to the formula : AC = \/20 2 - 12 2 = V400- 144 = V^56 =16. Assume that AB = 15 and AC = 20. Find BC. BC =Vl5 2 + 20 2 = V 225 + 400 = \/625 = 25. The rules and formulas given make it possible to find the length of the sides in a right-angled tri- angle. To -find the angles, however, use must be ELEMENTS OF TRIGONOMETRY 101 made of the trigonometric functions, the meanings of which will be presently explained. The trigo- nometric functions are the sine, cosine, tangent, co- tangent, secant and cosecant of angles. While these functions are used in the solution of all kinds of triangles, they refer directly to right-angled tri- angles, and the meaning or value of each function can be explained by reference to a right-angled triangle as shown in Fig. 91, at G, where the side BC is the hypotenuse, AC the side adjacent to angle D, and AB the side opposite angle D. Of course, if reference is made to angle E, then AB is the side adjacent and AC the side opposite. The sine of an angle is the length of the opposite side, if the hypotenuse is assumed to equal 1. The sine of angle D, then, is the length of AB if BC equals 1. To find the sine of D when BC is any other length, divide AB by the length of BC. To find the sine of D, if BC equals 5, for example, it is necessary to divide the length of AB by 5. Find the sine of D, when AB = 15 and BC = 20. The sine of D = 15 * 20 = 0.75. The cosine of an angle is the length of the adja- cent side, if the hypotenuse is assumed to equal 1. The cosine of angle D, then, is the length of AC if BC equals 1. To find the cosine of D when BC is any other length, divide -AC by the length of BC. To find the cosine of D, if BC equals 8, for example, it is necessary to divide the length of AC by 8. Find the cosine of D, when AC = 12 and BC = 30. The cosine of D = 12 -*- 30 = 0.4. The tangent of an angle is the length of th< 102 SELF-TAUGHT MECHANICAL DRAWING posite side, if the adjacent side is assumed to equal 1. The tangent of angle D is the length of AB if AC equals 1. To find the tangent of D when AC equals any other length, divide AB by the length of AC. To find the tangent of D when AC equals 3, for example, it is necessary to divide the length of AB by 3. Find the tangent of D, when AB = 16 and AC = 12. The tangent of D = 16 + 12 = 1.333. The cotangent of an angle is the length of the adjacent side, if the opposite side is assumed to equal 1. The cotangent of angle D is the length of AC if AB equals 1. To find the cotangent of D when AB equals any other length, divide AC by the length of AB. To find the cotangent of D when AB equals 12, for example, divide AC by 12. Find the cotangent of D when AB = 3 and AC = 36. The cotangent of D = 36 + 3 = 12. The secant of an angle is the length of the hypo- tenuse, if the adjacent side is assumed to equal 1. The secant of angle D is the length of BC when AC equals 1. To find the secant of -D when AC is any other length, divide BC by the length of AC. Find the secant of D when BC = 24 and AC = 9. The secant of D = 24 -*- 9 = 2.666. . . The cosecant of an angle is the length of the hypotenuse if the opposite side is assumed to equal 1. The cosecant of angle D is the length of BC when AB equals 1. To find the cosecant of D when AB is any other length, divide BC by the length of AB. Find the cosecant of D when BC = 30 and AB = 3.75. The cosecant of D = 30 ^ 3.75 = 8. ELEMENTS OF TRIGONOMETRY 103 The expressions sine, cosine, tangent, cotangent, secant and cosecant are abbreviated as follows: sin, cos, tan, cot, sec, and cosec. Instead of writ- ing tangent of D, for example, it is usual to write tan D. By means of these functions, tables of which are given in the following, the values of angles can be introduced in the calculations of tri- angles. The tables here used give the values of the functions of angles for every degree and for every ten minutes. Only three decimal places are given, as that is enough for the great majority of shop calculations. When very accurate calculations are required, tables can be procured giving the functions for every minute, and with five decimal places. From the tables given, when the angle is known, the corresponding angular function can be found, and when the function is known, the cor- responding angle can be determined by merely reading off the values in the table. The tables in- clude sines, cosines, tangents and cotangents only, as these are most commonly used, and all problems can be solved by the use of them. When the se- cant is required, it can be found by dividing 1 by the cosine. The cosecant is found by dividing 1 by the sine. The tables of sines, cosines, etc., are read the same as any other table. It will be seen that the four tables given are headed Sines, Cosines, Tan- gents, and Cotangents, respectively. At the bottom of the table headed "Sines" is read the word "Cosines," and at the bottom of the table headed "Cosines" is read the word "Sines." In the same way, at the bottom of the table headed "Tan- 104 SINES MINUTES. DEG. DEG. 0' 10' 20' 30' 40' 50' 60' 0.000 0.003 0.006 0.009 0.012 0.015 0.017 89 1 0.017 0.020 0.023 0.026 0.029 0.032 0.035 88 2 0.035 0.038 0.041 0.044 0.047 0.049 0.052 87 3 0.052 0.055 0.058 0.061 0.064 0.067 0.070 86 4 0.070 0.073 0.076 0.078 0.081 0.084 0.087 85 5 0.087 0.090 0.093 0.096 0.099 0.102 0.105 84 6 0.105 0.107 0.110 0.113 0.116 0.119 0.122 83 7 0.122 0.125 0.128 0.131 0.133 0.136 0.139 82 8 0.139 0.142 0.145 0.148 0.151 0.154 0.156 81 9 0.156 0.159 0.162 0.165 0.168 0.171 0.174 80 10 0.174 0.177 0.179 0.182 0.185 0.188 0.191 79 11 0.191 0.194 0.197 0.199 0.202 0.205 0.208 78 12 0.208 0.211 0.214 0.216 0.219 0.222 0.225 "77 13 0.225 0.228 0.231 0.233 0.236 0.239 0.242 76 14 0.242 0.245 0.248 0.250 0.253 0.256 0.259 75 15 0.259 0.262 0.264 0.267 0.270 0.273 0.276 74 16 0.276 0.278 0.281 0.284 0.287 0.290 0.292 73 17 0.292 0.295 0.298 0.301 0.303 0.306 0.309 72 18 0.309 0.312 0.315 0.317 0.320 0.323 0.326 71 19 0.326 0.328 0.331 0.334 0.337 0.339 0.342 70 20 0.342 0.345 0.347 0.350 0.353 0.35) 0.358 69 21 0.358 0.361 0.364 0.367 0.369 0.372 0.375 68 22 0.375 0.377 0.380 0.383 0.385 0.388 0.391 67 23 0.391 0.393 0.396 0.399 0.401 0.404 0.407 66 24 0.407 0.409 0.412 0.415 0.417 0.420 0.423 65 25 0.423" 0.425 0.428 0.431 0.433 0.436 0.438 64 26 0.438 0.441 0.444 0.446 0.449 0.451 0.454 63 27 0.454 0.457 0.459 0.482 0.464 0.467 0.469 62 28 0.469 0.472 0.475 0.477 0.480 0.482 0.485 61 29 0.485 0.487 0.490 0.492 0.495 0.497 0.500 60 30 0.500 0.503 0.505 0.508 0.510 0,513 0.515 59 31 0.515 0.518 0.520 0.522 0.525 0.527 0.530 58 32 0.530 0.532 0.535 0.537 0.540 0.542 0.545 57 33 0.545 0.547 0.553 0.552 0.554 0.557 0.559 56 34 0.559 0.562 0.564 0.566 0.569 0.571 0.574 55 35 0.574 0.576 0.578 0.581 0.583 0.585 0.588 54 36 0.588 0.590 0.592 0.595 0.597 0.599 0.602 53 37 0.602 0.604 0.606 0.609 0.611 0.613 0.616 52 38 0.616 0.618 0.620 0.623 0.625 0.627 0.629 51 39 0.629 0.632 0.634 0.636 0.638 0.641 0.643 50 40 0.643 0.645 0.647 0.649 0.652 0.654 0.656 49 41 0.656 0.658 0.660 0.663 0.665 0.667 0.669 48 42 0.669 0.671 0.673 0.676 0.678 0.680 0.682 47 43 0.682 0.684 0.686 0.688 0.690 0.693 0.695 46 44 0.695 0.697 0.699 0.701 0.703 0.705 0.707 45 60' 50' 40' 30' 20' 10' 0' MINUTES. COSINES COSINES 105 MINUTES. DEC. DEG. 0' 10' 20' 30' 40' 50' 60' 1.000 1.000 1.000 1.000 1.000 1.000 1.000 ~~89~~ 1 1.000 1.000 1.000 1.000 1.000 0.999 0.999 88 2 0.999 0.999 0.999 0.999 0.999 0.999 0.999 87 3 0.999 0.998 0.998 0.998 0.998 0.998 0.998 86 4 0.998 0.997 0.997 0.997 0.997 0.996 0.996 85 5 0.996 0.996 0.993 0.995 0.995 0.995 0.995 84 6 0.995 0.994 0.994 0.994 0.993 0.993 0.993 83 7 0.993 0.992 0.992 0.991 0.991 0.991 0.990 82 8 0.990 0.990 0.989 0.989 0.98) 0.988 0.988 81 9 0.988 0.987 0.987 0.986 0.986 0.985 0.985 80 10 0.985 0.984 0.984 0.983 0.983 0.982 0.982 79 11 0.982 0.981 0.981 0.980 0.979 0.979 0.978 78 12 0.978 0.978 0.977 0.976 0.976 0.975 0.974 77 13 0.974 0.974 0.973 0.972 0.972 "0.971 0.970 76 14 0.970 0.970 0.969 0.968 0.967 0.967 0.966 75 15 0.966 0.965 0.964 0.964 0.963 0.962 0.961 . 74 16 0.961 0.960 0.960 0.959 0.958 0.957 0.956 73 17 0.956 0.955 0.955 0.954 0.953 0.952 0.951 72 18 0.951 0.950 0.949 0.948 0.947 0.946 0.946 71 19 0.946 0.945 0.944 0.943 0.942 0.941 0.940 70 20 0.940 0.939 0.938 0.937 0.936 0.935 0.934 69 21 0.934 0.933 0.931 0.930 0.929 0.928 0.927 68 22 0.927 0.926 0.925 0.924 0.923 0.922 0.921 67 23 0.921 0.919 0.918 0.917 0.916 0.915 0.914 66 24 0.914 0.912 0.911 0.910 0.909 0.908 0.906 65 25 0.906 0.905 0.904 0.903 0.901 0.900 0.899 64 26 0.899 0.898 0.896 0.895 0.894 0.892 0.891 63 27 0.891 0.890 0.888 0.887 0.886 0.884 0.883 62 28 0.883 0.882 0.880 0.879 0.877 0.876 0.875 61 29 0.875 0.873 0.872 0.870 0.869 0.867 0.866 60 30 0.866 0.865 0.863 0.862 0.860 0.859 0.857 59 31 0.857 0.856 0.854 0.853 0.851 0.850 0.848 58 32 0.848 0.847 0.845 0.843 0.842 0.840 0.839 57 33 0.839 0.837 0.835 0.834 0.832 0.831 0.829 56 34 0.829 0.827 0.826 0.824 0.822 0.821 0.819 55 35 0.819 0.817 0.816 0.814 0.812 0.811 0.809 54 36 0.809 0.807 0.806 0.804 0.802 0.800 0.799 53 37 0.799 0.797 0.795 0.793 0.792 0.790 0.788 52 38 0.788 0.786 0.784 0.783 0.781 0.779 0.777 51 39 0.777 0.775 0.773 0.772 0.770 0.768 0.766 50 40 0.766 0.764 0.762 0.760 0.759 0.757 0.755 49 41 0.755 0.753 0.751 0.749 0.747 0.745 0.743 48 42 0.743 0.741 0.739 0.737 0.735 0.733 0.731 47 43 0.731 0.729 0.727 0.725 0.723 0.721 0.719 46 44 0.719 0.717 0.715 0.713 0.711 0.709 0.707 45 T^FT 1 60' 50' 40' 30' 20' 10' 0' J-^-h-O . MINUTES. DEG. SINES 106 TANGENTS DEG. MINUTES. DEG. 0' 10' 20' 30' 40' 50' 60' 0.000 0.003 0.006 0.009 0.012 0.015 0.017 89 1 0.017 0.020 0.023 0.026 0.029 0.032 0.035 88 2 0.035 0.038 0.041 0.044 0.047 0.049 0.02 87 3 0.052 0.055 0.058 0.061 0.064 0.067 0.070 86 4 0.070 0.073 0.076 0.079 0.082 0.085 O.C87 85 5 0.087 0.090 0.093 0.096 0.099 0.102 0.105 84 6 0.105 0.108 0.111 0.114 0.117 0.120 0.123 83 7 0.123 0.126 0.129 0.132 0.135 0.138 0.141 82 8 0.141 0.144 0.146 0.149 0.152 0.155 o.l8 81 9 0.158 0.161 0.164 0.167 0.170 0.173 0.176 80 10 0.176 0.179 0.182 0.185 0.188 0.191 0.194 79 11 0.194 0.197 0.200 0.203 0.206 0.210 0.213 78 12 0.213 0.216 0.219 0.222 0.225 0.228 0.231 77 13 0.231 0.234 0.237 0.240 0.243 0.246 0.249 76 14 0.249 0.252 0.256 0.259 0.262 0.265 0.268 75 15 0.268 0.271 0.274 0.277 0.280 0.284 0.287 74 16 0.287 0.290 0.293 0.296 0.299 0.303 0.306 73 17 0.306 0.309 0.312 0.315 0.318 0.322 0.325 72 18 0.325 0.328 0.331 0.335 0.338 0.341 0.344 71 19 0.344 0.348 0.351 0.354 0.357 0.361 0.364 70 20 0.364 0.367 0.371 0.374 0.377 0.381 0.384 69 21 0.384 0.387 0.391 0.394 0.397 0.401 0.404 68 22 0.404 0.407 0.411 0.414 0.418 0.421 0.424 67 23 0.424 0.428 0.431 0.435 0.438 0.442 0.445 66 24 0.445 0.449 0.452 0.456 0.459 0.463 0.466 65 25 0.466 0.470 0.473 0.477 0.481 0.484 0.488 64 26 0.488 0.491 0.495 0.499 0.502 0.506 0.510 63 27 0.510 0.513 0.517 0.521 0.524 0.528 0.532 62 28 0.532 0.535 0.539 0.543 0.547 0.551 0.554 61 29 0.554 0.558 0.562 0.566 0.570 0.573 0.577 60 30 0.577 0.581 0.585 0.589 0.593 0.597 0.601 59 31 0.601 0.605 0.609 0.613 0.617 0.621 0.625 58 32 0.625 0.629 0.633 0.637 0.641 0.645 0.649 57 33 0.649 0.654 0.658 0.662 0.666 0.670 0.675 56 34 0.675 0.679 0.683 0.687 0.692 0.696 0.700 55 35 0.700 0.705 0.709 0.713 0.718 0.722 0.727 54 36 0.727 0.731 0.735 0.740 0.744 0.749 0.754 53 37 0.754 0.758 0.763 0.767 0.772 0.777 0.781 52 38 0.781 0.786 0.791 0.795 0.800 0.805 0.810 51 39 0.810 0.815 0. 819 ! 0.824 0.829 0.834 0.839 50 40 0.839 0.844 0.849 0.854 0.859 0.864 0.869 49 41 0.869 0.874 0.880 0.885 0.890 0.895 0.900 48 42 0.900 0.906 0.911 0.916 0.922 0.927 0.933 47 43 0.933 0.938 0.943 0.949 0.955 0.960 0.966 46 44 0.966 0.971 0.977 0.983 0.988 0.994 1.000 45 60' 50' 40' 30' 20' 10' 0' DEG. MINUTES. DEG. COTANGENTS COTANGENTS 107 MINUTES. DEC. DEG. 0' 10' 20' 30' 40' 50' 60' oo 343.8 171.9 114.6 85.94 68.75 57.29 89 1 57. 29 49.10 42.96 38.19 34.37 31.24 28.64 88 2 28.64 26.43 24 . 54 22.90 21.47 20.21 19.08 87 3 19.08 18.07 17.17 16.35 15.60 14.92 14.30 86 4 14.30 13.73 13.20 12.71 12.25 11.83 11.43 85 5 11.43 11.06 10.71 10.39 10.08 9.788 9.514 84 6 9.514 9.225 9.010 8.777 8.556 8.345 8.144 83 7 8.144 7.953 7.770 7.596 7.429 7.269 7.115 82 8 7.115 6.968 6.827 6.691 6.561 6.435 6.314 81 9 6.314 6.197 6.084 5.976 5.871 5.769 5.671 80 10 5.671 5.576 5.485 5.396 5.309 5.226 5.145 79 11 5.145 5.066 4.989 4.915 4.843 4.773 4.705 78 12 4.705 4.638 4.574 4.511 4.449 4.390 4.331 77 13 4.331 4.275 4,219 4.165 4.113 4.061 4.011 76 14 4.011 3.962 3.914 3.867 3.821 3.776 3.732 75 15 3.732 3.689 3.647 3.606 3.566 3.526 3.487 74 16 3.487 3.450 3.412 3.376 3.340 3.305 3.271 73 17 3.271 3.237 3.204 3.172 3.140 3.108 3.078 72 18 3.078 3.047 3.018 2.989 2.960 2.932 2.904 71 19 2.904 2.877 2.850 2.824 2.798 2.773 2.747 70 20 2.747 2.723 2.699 2.675 2.651 2.628 2.605 69 21 2.605 2.583 2.560 2.539 2.517 2.496 2.475 68 22 2.475 2.455 2.434 2.414 2.394 2.375 2.356 67 23 2.356 2.337 2.318 2.300 2.282 2.264 2.246 66 24 2.246 2.229 2.211 2.194 2.177 2.161 2.145 65 25 2.145 2.128 2.112 2.097 2.081 2.066 2.050 64 26 2.050 2.035 2.020 2.006 1.991 1.977 1.963 63 27 .963 1.949 1.935 1.921 1.907 1.894 1.881 62 28 .881 1.868 1.855 1.842 1.829 1.816 1.804 61 29 .804 1.792 1.780 1.767 1.756 1.744 1.732 60 30 .732 1.720 1.709 1.698 1.686 1.675 1.664 59 31 .664 1.653 1.643 .632 1.621 1.611 1.600 58 32 .600 1.590 1.580 .570 .560 1.550 1.540 57 33 1.540 1.530 1.520 .511 .501 1.492 1.483 56 34 1.483 1.473 1.464 .455 .446 1.437 1.428 55 35 1.428 1.419 .411 .402 .393 1.385 1.376 54 36 1.376 1.368 .360 .351 .343 1.335 1.327 53 37 1.327 1.319 .311 .303 .295 1.288 1.280 52 38 1.280 1.272 .265 .257 .250 1.242 1.235 51 39 .235 1.228 1.220 .213 .206 1.199 1.192 50 40 .192 1.185 .178 .171 .164 1.157 1.150 49 41 .150 1.144 1.137 .130 .124 1.117 1.111 48 42 .111 1.104 .098 .091 .085 1.079 1.072 47 43 .072 1.066 .060 .054 .048 1.042 1.036 46 44 .036 1.030 .024 .018 .012 1.006 1.000 45 60' 50' 40' 30' 20' 10' O' DTTP T^T1/~l -L/rj(j . MINUTES. J_JEG. TANGENTS 108 SELF-TAUGHT MECHANICAL DRAWING gents, " we read "Cotangents/' and at the bottom of the table headed " Cotangents, " we read " Tan- gents/' The object of this will be presently ex- plained. The extreme left-hand column, we find, is headed "Deg.," and the following seven columns are headed 0', 10', 20', 30', 40', 50' and 60', re- spectively, these columns indicating the minutes. At the bottom of the pages the same numbers are found but reading from the right to the left. The values of the functions marked at the top are read in the table opposite the degrees in the left-hand column and under the minutes at top. The values of the functions marked at the bottom are read opposite the degrees in the right-hand column and over the minutes at the bottom. For example, the sine of 39 40 ' or sin 39 40 ', as it is written in formulas, is thus found to be 0.638, and the sine of 64 10' is 0.900, this latter value being read off in the second table, reading it from the bottom up, and locating the number of degrees in the right- hand column. As further examples, we find tan 37 40 ' = 0.772 cot 37 40 ' = 1.295 tan80 0' = 5.671 cos 75 30 ' = 0.250 We are now ready to proceed to solve right-ang- led triangles with regard both to the sides and the angles. In any right-angled triangle, if either two sides, or one side and one of the acute angles are known, the remaining quantities can be found. As a general rule, in any triangle, all the quantities ELEMENTS OF TRIGONOMETRY 109 can be found when three quantities, at least one of which is a side, are given. In a right-angled triangle the right angle is always known, of course, so that here, therefore, only two additional quantities are necessary. If all the three angles are known, the length of the sides cannot be de- termined; one side, at least, must also always be known in order to make possible the solution of the triangle. The following rules should be used for solving right-angled triangles. Case 1. Two sides known. Use the rules al- ready given in this chapter for finding the third T i r ' 1 GIVEN ANGL < ADJACENT SIDE >j FIG. 92. side when two sides in a right-angled triangle are given. To find the angles use the rules already given for finding sines, cosines, etc., and the tables. Case 2. Hypotenuse and one angle given. Call the side adjacent to the given angle the adjacent side, and the side opposite the given angle the opposite side (see Fig. 92.) Then the adjacent side equals the hypotenuse multiplied by the cosine 110 SELF-TAUGHT MECHANICAL DRAWING of the given angle; the opposite side equals the hypotenuse multiplied by the sine of the given angle ; and the unknown angle equals 90 degrees minus the given angle. Case 3. One angle and its adjacent side given. The hypotenuse equals the adjacent side divided by the cosine of the given angle ; the opposite side equals the adjacent side multiplied by the tangent of the given angle; and the unknown angle is found as in Case 2. Case 4. One angle and its opposite side known. The hypotenuse equals the opposite side divided by the sine of the given angle; the adjacent side equals the opposite side multiplied by the cotangent of the given angle; and the unknown angle is found as in Case 2. These rules may be written as formulas as fol- lows (see Fig. 93) : Case 1. For formulas for the sides see the first part of this Chapter. For the angles we have: sin B = - sin C = . a a Case 2. Here, when a and B are given, we have : c = a cos B; b = a sin B; C = 90 - B. When a and C are given, we have : b = a cos C; c = a sin C; = 90 - C. Case 3. Here, when B and c are given, we have: a = C -^; b = c tan B; C = 90 - B. cos B When C and 6 are given, we have: a = ~; c = 6 tan C; B - 90 - C. cos C' ELEMENTS OF TRIGONOMETRY 111 Case 4. Here, when B and b are known, we have: ,; c = b cot B; C = 90 - B. sin When C and c are known, we have : a - ~\ 6 - c cot C; sin o 90 - C. These rules and formulas, while not including all possible combinations for the solution of right- angled triangles, give all the information neces- sary for the solution of any kind of a right-angled A C Bf r- FIG. 94. FIG. 95. triangle. A few examples of the use of these rules and formulas will now be given, so as to clearly indicate the mode of procedure in practical work. Example 1. In the triangle in Fig. 94, side A C is 12 inches long and angle D is 40 degrees. Find angle E and the two unknown sides. This is an example of Case 3, one angle and its adjacent side being given. Angle E equals 90 de- grees minus the given angle, or #=90 -40 -50 112 SELF-TAUGHT MECHANICAL DRAWING The hypotenuse BC equals the adjacent side divided by the cosine of D, or BC - - = 15 - 666 inches - Side AB equals the adjacent side multiplied by the tangent of D, or AB = 12 X tan 40 = 12 X 0.839 = 10.068 inches. The cosine and tangent of 40 degrees are found in the tables of trigonometric functions as already explained. Example 2. In the triangle in Fig. 95, the hypotenuse BC = 17i inches. One angle is 44 de- grees. Find angle E and the sides AB and AC. This is an example of Case 2, the hypotenuse and one 'angle being given. Using the rules or formulas given for Case 2, we have : AC = 174 X cos 44 = 17.5 X 0.719 = 12.5825 inches. AB = 174 X sin 44 = 17.5 X 0.695 = 12.1625 inches. E =90 -44 =46. Example 3. In the triangle in Fig. 96, side AC = 208 feet, and the angle opposite this side = 38 degrees. Find angle E, and the two remaining sides. This is an example of Case 4, one side and the angle opposite it being known. From the rules or formulas given for Case 4, we have: BC = 208 - sin 38 = 208 - 0.616 = 337.66 feet. AB = 208 X cot 38 = 208 X 1.280 = 266.24 feet. #=90 -38 = 52. ELEMENTS OF TRIGONOMETRY 113 Example 4. In the triangle in Fig. 97, side AC = 3 inches, and the hypotenuse #C=5 inches. Find side AB and angles D and E. This is an example of Case 1. According to a formula previously given in this chapter AB = VBC' 2 - AC 2 = Vl6 = 4. AB sin E = BC \/5 2 - 3 2 = \/25 - 9 = =0.800. From the tables we find that the angle corre- FIG. 96. FIG. 97. spending to a sine which equals 0.800 is 53 10'. Consequently : # = 53 10', and D = 90 -53 10' = 36 50'. Example 5. In the triangle in Fig. 98, side BC, the hypotenuse, is 1| inch long. One angle is 65 degrees. Find angle E and the remaining sides. 114 SELF-TAUGHT MECHANICAL DRAWING This is an example of Case 2. We have: E= 90 -65 =25. AB= 1| X cos 65 = 1.375 X 0.423 - 0.5816 inch. AC = 1| X sin 65 = 1.375 X 0.906 =1.2457 inch. Example 6. In the triangle in Fig. 99, side AB = 0.706 inch, and the angle adjacent to this side is 60 degrees. Find angle E and the sides AC and EC. A B T U ia FIG. 99. This is an example of Case 3. We have : #=90 -60 = 30. EC = 0.706 *- cos 60 - 0.706 - 0.500 = 1.412 inch. AC= 0.706 X tan 60 = 0.706 X 1.732= 1.2228 inch. The previous examples, carefully studied, will give a comprehensive idea of the methods used for solving right-angled triangles, no matter which parts are given or unknown. A triangle which does not contain a right angle is called an oblique triangle. Any such triangle can be solved by the aid of the formulas given for the right triangle, by dividing it into two right- angled triangles by means of a line drawn from the vertex of one angle perpendicular towards the opposite side. Formulas can be deduced which do ELEMENTS OF TRIGONOMETRY 115 not require that the triangle be so divided, but for elementary purposes, the method indicated is the most easily understood. In Fig. 100, for example, a triangle is given as shown. One angle is 50 degrees, and the sides in- cluding this angle are 4 and 5 inches long, respec- tively. Draw a line from A perpendicular to the side EC. We have here two right-angled tri- angles, and can now proceed by using the formulas previously given. In triangle ADB, the hypoten- use AB and one angle are given. We then find side AD by means of the formulas for Case 2, and also angle BAD and side BD. Next we find CD = 5- BD. We then, in the triangle A CD know two sides AD and CD, and can thus find side AC as in Case 1, as well as angles A CD and CAD. The angle BAG finally is found by adding angles 116 SELF-TAUGHT MECHANICAL DRAWING BAD and CAD and, then, all the angles and sides in the triangle are found. The successive calculations would be carried out as follows: AD = 4 X sin 50 = 4 X 0.766 - 3.064. D = 4 X cos 50= 4 X 0.643 = 2.572. Angle BAD = 90 - 50 = 40. DC = 5 - BD = 5 - 2.572 = 2.428. AC = AD* + DC* = aoe 2.428* =- 3.91. Sine of angle ACD = ~ = ~ - 0.784. Angle A CD = 51 40'. Angle CAD = 90- 51 40' = 38 20'. Angle BAG = 40 + 38 20'= 78 20'. In order to check the results obtained, add angles ABC, BAG and ACD. The sum of these angles must equal 180 degrees if the results are correct: 50 + 78 20' + 51 40' = 180. This method, with such modifications as are necessary to meet the different requirements in each problem, may be used for solving all oblique- angled triangles, except in the case where no angle is known, but only the lengths of all the three sides. In this case the use of a direct formula will prove the best and most convenient. Let the three known sides be a, b and c, and the angles opposite each of them A, B and C, respectively, as in Fig. 101 ; then we have : b 2 + c 2 - a 2 b sin A 180- (A + B). ELEMENTS OF TRIGONOMETRY 117 As an example, assume that the three sides in a triangle are a = 4, 6 = 5, and c = 6 inches long. Find the angles. 5 2 + 6 2 - 4 2 = 45_ 60 Cos 4 = 2X5X6 ,4 = 41 25'. 0.750. Sin B = _ 4 4 B = 55 50'. C = 180 - (41 25' + 55 50') = 82 45'. As only the first principles of trigonometry have here been treated, some of the more advanced ^ problems have, by necessity, been omitted. For ordinary shop calculations the present treatment will, however, be found more satisfactory, as some of the matter which would unnecessarily burden the mind has been left out. If the student only first acquires a thorough understanding of the first 118 SELF-TAUGHT MECHANICAL DRAWING principles of mathematics and their application to machine design, it is comparatively easy to broaden the field of one's knowledge; it is, therefore, of extreme importance that these first principles be thoroughly understood and digested. The ap- plication will then be found comparatively easy. The trigonometric functions afford a convenient means for laying out angles ; and when the sides A r r 60 >j ^FiG. 102. Method of Laying Out Angles by Means of Natural Functions. of the angle laid out are much extended, it can be laid out more accurately in this manner than by the use of an ordinary protractor. Let it be required, for instance, to lay out an angle of 37 degrees, one side of the angle being 60 inches long. Lay out the side AB, Fig. 102, 60 inches long. Then with a radius equal to the sine of 37 degrees multiplied by 60, and with a center at B, draw an ELEMENTS OF TRIGONOMETRY 119 arc C. Then draw a line from A, tangent to arc C. This line forms an angle of 37 degrees with line AB. If the required angle is over 45 degrees, then it is preferable to lay out the complement angle from a line perpendicular to the original -B FIG- 103. Laying Out an Angle Greater than 45 Degrees. line, as shown in Fig. 103, where an angle of 70 degrees is to be laid out, but the 20-degree comple- ment angle is actually constructed. Many other methods for use in laying out angles, arcs, etc., will readily suggest themselves to the student who thoroughly understands the relation of the trigo- nometric functions in a right-angled triangle. CHAPTER VIII ELEMENTS OF MECHANICS MECHANICS is defined as that science, or branch of applied mathematics, which treats of the action of forces on bodies. That part of mechanics which considers the action of forces in producing rest or equilibrium is called statics; that which relates to such action in producing motion is called dynamics; the term mechanics includes the action of forces on all bodies whether solid, liquid or gaseous. It is sometimes, however, and formerly was often, used distinctively of solid bodies only. The me- chanics of liquid bodies is called also hydrostatics or hydrodynamics, according as the laws of rest or motion are considered. The mechanics of gaseous bodies is called also pneumatics. The mechanics of fluids in motion, with special reference to the methods of obtaining from them useful results, constitutes hydraulics. The Resultant of Two or More Forces. When a body is acted upon by several forces of different magnitudes in different directions, a single force may be found, which in direction and magnitude will be a resultant of the action of the several forces. The magnitude and direction of this single force may be obtained by what is known as the parallelogram of forces. Let A and B, Fig. 104, 120 ELEMENTS OF MECHANICS 121 represent the direction of two forces acting simul- taneously upon P, and let their lengths represent the relative magnitude of the forces ; then, to find a force which in direction and magnitude shall be a resultant of these two forces, draw the line C parallel with B, and draw the line D parallel with A. A diagonal of the parallelogram thus formed, drawn from Pto E, will give the direction, and its F FIG. 104. Parallelogram of Forces. length as compared with A and B, the relative magnitude, of the required force. That this is so may be seen by considering the two forces as acting separately upon P. Let A be considered as acting upon P to move it through a distance equal to its length. Then P would be moved to F. If the force B is now caused to act upon P to move it through a distance equal to its length, P will arrive at G. As FP has the same length and direction as A, and as GFhas the same length and direction as B, the distance from G to P would be the same as the distance from P to E; therefore, PE, the diagonal of the parallelogram formed by the lines A, B, C, and D, represents the required new force or resultant. If there are more than two forces acting upon the point P, first find a resultant of any two of the forces; then consider this resultant as replacing 122 SELF-TAUGHT MECHANICAL DRAWING FIG. 105. -Resultant of Three Forces. the first two, and find the resultant of it and an- other of the original forces ; continue this process until a force is obtained which will be the resultant of all of the original forces. Thus, in Fig. 105, if A, B and C be considered as representing in di- rection and magnitude three forces which are acting simultaneously uponP; then, if we draw a parallelogram upon A and B, we have its diag- onal PD as the resultant of A and B. A parallel- ogram is now drawn upon PD and C, giving PE, its diagonal, as the resultant of these two, and, consequently, of the three original forces. This principle holds true whether the original forces are acting in the same plane or not. Thus, in Fig. 106, let A, B and C be three forces acting simultaneously upon P. Then the re- sultant of A and B would be the diagonal PD. Considering this as replacing A and J5, a resultant of it and C would be a diagonal drawn from P to the further corner E; PE would then be the resultant of A, B andC. This operation may, of course, be reversed to allow of finding two or more forces in different FIG. 106. -Resultant of Three Forces in Different Planes. ELEMENTS OF MECHANICS 123 directions which in magnitude shall be equivalent to a single known force. Thus in Fig. 107, if PA represents the direction and magnitude of a given force which it is desired to replace by two others acting in the direction of PB and PC, respectively, then draw a line from A to PB parallel with PC, and draw another from A to PC parallel with PB. The lengths Pa and Pb thus determined will represent the relative magnitudes, as com- pared with PA, of the required new forces. Parallel Forces. Let A and B, Fig. 108, represent the direction and magnitude of two parallel forces acting together upon the bar DE. These two forces may be replaced or counterbalanced by a single force, equal in magnitude to A and B combined. To de- termine the point of application of this new force produce A to a, making Da equal in length to B. Also make bE equal in length to A. The inter- section of the line connecting a and b with DE, at F, will be the required point of application. The lengths DF and FE will be inversely proportional to the forces A and B. That is, the length FE will be to the force A as the length DF is to the force B. The product of DF multiplied by A will be equal to the product of FE multiplied by B. Fig. 109 shows how several parallel forces, act- ing in the same direction, may be replaced or FIG. 107. Resolution of Forces. 124 SELF-TAUGHT MECHANICAL DRAWING counterbalanced by a single force. Let A, B and C represent the relative magnitudes of the forces. A resultant of B and C would be D, equal in / xi. FIG. 108. Parallel Forces. FIG. 109. Resultant of Several Parallel Forces. magnitude to B and C combined, and its point of application, determined in the manner previously described, would be at a. Regarding D as a single force replacing B and C, would give E, equal in magnitude to A and D combined, as the result- ant of these two, and its point of application, de- termined as before, would be at b. Oblique Forces. Let A and B, Fig. 110, repre- sent the directions and relative magnitudes of two forces acting simultaneously upon the barDE. These two forces may be either replaced or counter- FIG. 110. Oblique- Forces Acting at Different Points on a Bar. ELEMENTS OF MECHANICS 125 balanced by a single force, which in direction and magnitude shall be a resultant of them. Produce A and B until they meet at a. Draw the parallel- ogram abed, making da equal to A, and ba equal to B. The diagonal of this parallelogram will give the direction and relative magnitude of the new force, and if extended its intersection with DE will give the point of application. Opposing Forces. Let A and J9, Fig. Ill, repre- sent the directions and relative magnitudes of two forces acting upon oppo- site sides of the bar DE. These two forces may be replaced by a single force, which in direction and magnitude will be a re- sultant of them. Produce A and B until they meet at a. Lay off ac equal to the length of B, and make be equal to and parallel with A. A line drawn from a to 6 will give the direction of the new force, and the length of ab, as compared with A and B will give its relative magnitude. Its application on bar DE may be de- termined by extending ab until it intersects DE. Levers. When a workman wishes to raise a heavy object, he may insert one end of a bar un- der it, and lift on the other end; or, pushing a block of wood or iron in under the bar as close to the object to be raised as he can, he presses down upon the free end of the bar. A bar so used con- FIG. 111. Opposing Oblique Forces. 126 SELF-TAUGHT MECHANICAL DRAWING stitutes a lever, and the point where the bar rests when the lever is doing its work, the end of the bar in under the heavy object in the first case, or the block on which the bar rests in the second case, is the fulcrum of the lever. Levers are of three kinds, as shown in Fig. 112: First, where the fulcrum is between the power J1ST I 3RD O w FIG. 112. Classes of Levers. and the weight; second, where the weight is between the fulcrum and the power; and, third, where the power is between the fulcrum and the weight. A man's forearm furnishes a good illus- tration of a lever of the third class, the fulcrum being at the elbow, the weight at the hand, and the muscle, being attached to the bone of the arm, at a short distance from the elbow, furnishing the power. ELEMENTS OF MECHANICS 127 In all of these cases the gain in power is exactly proportional to the loss in speed, or the gain in speed is exactly proportional to the loss in power. Also, in every case the product of the weight mul- tiplied by its distance from the fulcrum, will equal the product of the power multiplied by its distance from the fulcrum, or, the weight and power will balance each other when the weight multiplied by the distance through which it moves, equals the power multiplied by the distance through which it moves. If in Fig. 108 the bar DE is a lever, the fulcrum will be at F, and the methods used in that figure and in Figs. 109, 110 and 111 give solutions of dif- ferent lever problems. The length of the lever arm is independent of the form of the lever. In Fig. 113 is shown a lever FIG. 113. -Lever of Curved Shape. of curved shape ; but the lever arms on which the calculation as to the work that the lever is doing, will be based, will be straight lines connecting the point where the power is applied, or the point which supports the weight, with the fulcrum. The length of the lever arm is always at right angles to the direction in which the power is being 128 SELF-TAUGHT MECHANICAL DRAWING applied, or to the direction of the resistance of the weight or load. In Fig. 114 two cases are shown where the power is applied obliquely on the lever; but the lever arm on which the calculation is based will be the dis- f\ FIG. 114. Power Applied Obliquely on Lever. tance Fa measured from the fulcrum, at right angles to the direction of the power. Compound Levers. In Fig. 115 is shown a case where the power gained with one lever is further increased by' the use of a second lever, acting on the first one. The weight and power will balance ELEMENTS OF MECHANICS 129 each other when the product of the weight and the lever arms ab and ef, multiplied together, equals the product of the power and the lever arms gfand be multiplied together. Thus, to find the weight 1 1 (_> ^F 9 V I I 1 e > FIG. 115. Compound Levers. which a given power will lift, divide the product of the power and its lever arms &/*and be, multi- plied together, by the product of the lever arms of the weight, ab and ef, multiplied together. To find 6W A B FIG. 116. Diagram for Lever Problem. the power necessary to lift a given weight, divide the product of the weight and its lever arms, ab and ef, multiplied together, by the product of the lever arms of the power, gf and be, multiplied together. 130 SELF-TAUGHT MECHANICAL DRAWING A few examples will illustrate these principles. Assume that in Fig. 116 a weight at A must bal- ance the 18-pound weight at B. The lever arms are given as 12 and 5 inches, respectively. How much must the weight W be, in order to balance the weight at B ? The weight at B (18 pounds) times its lever arm (5 inches) must equal the weight W times its lever arm (12 inches). In other words: 18 X 5 = W X 12. 90 = 12 W. W = 12 = 7i P unds - In Fig. 117, two weights, 4 and 2 pounds, respec- tively, are balanced by a weight W. Find what r ./ c FIG. 117. Diagram for Lever Problem. the weight of TFmust be with the lever arms given in the engraving. In this case the weight at A times its lever arm plus the weight at B times its lever arm, will equal weight W times its lever arm. The sum of the products of the weights and leverages of the weight at A and B is taken, because both these weights are on the same side of the fulcrum F. ELEMENTS OF MECHANICS 131 Carrying out the calculation outlined above, we have: 4 X 16 + 2X8 = 6 W. 64 + 16 = 80 = 6 W. W = 8 ^- = 134 pounds. b The product of a weight or force and its lever arm is commonly called the moment of the force. The moment of the force at A, for example, is 4 pounds X 16 inches = 64 inch-pounds. If the lever arm were 16 feet instead of 16 inches, the result would be 64 foot-pounds. An interesting application of the lever, and the moments of forces, is presented in calculations of FlG. 118. Diagram for Lever Problem. weights for safety valves. A diagrammatical sketch of a safety valve lever is shown in Fig. 118. Assume that the total steam pressure, acting on the whole area of the safety valve, is 300 pounds when it is required that the steam should "blow off." Find the weight W required near the end of the lever to keep the valve down until the total pressure is 300 pounds on the valve. Assume the weight of the lever itself to be 6 pounds, con- centrated at its center of gravity, 10 inches from the fulcrum F. 132 SELF-TAUGHT MECHANICAL DRAWING In this case we have that the moment of the steam pressure, which acts upward, should equal the sum of the moments of the weight of the lever and the weight W. Therefore : 300 X 3 = 6 X 10 + 20 W. 900 = 60 + 20 W. 900 - 60 = 20 W. 840 = 20 W. TJ7 840 , , W = -gQ- = 42 pounds. The calculation above has been carried out step by step, so that students unfamiliar with the alge- braic solution of equations may be able to under- stand the principles involved in simple examples of this kind. In the following, the calculations have been carried out more directly, but the stu- dent should use the "step by step" method until thoroughly familiar with the subject. Fixed and Movable Pulleys. A fixed pulley is frequently used to change the direction of the power, as shown in Fig. 119, but there is no gain in power with such a pulley, as there is no com- pensating loss of speed ; the weight will move up- ward at the same rate of speed as the power moves downward. If now a movable pulley be used in connection with the fixed pulley as shown in Fig. 120, then as the end of the rope to which the power is applied is drawn downward, each of the two strands of rope between the pulleys will take half of the stress of the suspended weight, and the weight will be raised only one-half the distance that the ELEMENTS OF MECHANICS 133 power descends. The power will therefore need to be only one-half of the weight. In Fig. 121, there are three strands of rope between the pulleys, each of which will be equally shortened when the free end of the rope is pulled ; the power, therefore, is only one-third of the weight. In Fig. 122, with o FIG. 119. Fixed Pulley. FIG. 120. Fixed and Movable Pulleys. four strands of rope between the pulleys, each fur- nishing an equal amount to the free end as it is drawn out, the power need be only one-fourth of the weight. The law of the pulley, then, where a single rope is employed, is that the power will be increased as many times as there are lines of rope between the pulleys to participate in the shortening. In a sys- tem using more than one rope, as shown in Fig. 134 SELF-TAUGHT MECHANICAL DRAWING 123, each additional movable pulley doubles the power, as it will move at only half the rate of the preceding pulley. Differential Pulleys. Another form of pulley, known as the differential pulley, much used in ma- chine shops, is shown in Fig. 124. In this form of w FIG. 121. Tackle where Load is Taken on Three Strands of Rope. FIG. 122. Tackle where Load is Taken on Four Strands of Rope. pulley an endless chain replaces the rope, the pul- leys themselves being grooved and toothed like sprocket wheels. The two pulleys at the top are of slightly different diameters, but rotate together as one piece. In operation, as the chain is drawn up by the large wheel it passes around in a loop to the small wheel from which it is unwound, causing the loop in which the movable pulley rests to be ELEMENTS OF MECHANICS 135 shortened by an amount equal to the difference in the pitch circumferences of the two upper wheels, when they have made one revolution. This would cause the weight to be raised one-half of that amount. If in a given case the two upper pulleys had respectively 20 and 19 teeth, then as the ap- FlG. 123. A Special Arrange- ment of Movable Pulleys. FIG. 124. Differential Pulley. plied power was being moved through a distance of 20 inches the small pulley would unwind 19 inches of the chain, causing a shortening of the loop in which the movable pulley rests of one inch, which would raise the weight one-half of an inch, giving a ratio of load to power of 40 to 1. In all of these cases the results actually attained in practice will be somewhat modified from the 136 SELF-TAUGHT MECHANICAL DRAWING theoretical results given by calculations, by the losses occasioned by friction. Inclined Planes. In raising heavy weights through short distances, as for instance in loading barrels onto wagons, a plank may be used to facili- tate the work by placing one end of it on the ground and the other end on the wagon, and roll- ing the barrel up the plank onto the wagon. Such an arrangement is called an inclined plane. When the force which is being applied to the rolling FIG. 125. Inclined Plane. FIG. 126. Power Applied Parallel to Base. object is exerted in a direction parallel to the in- clined surface, as in Fig. 125, it is evident that the power must move through a distance equal to the length of the incline in order to raise the weight the desired height. The gain in power will then be equal to the length of the incline divided by the height. If the power is applied in a direction parallel with the base, as in Fig. 126, the power will have to advance through a distance equal to the length of the base to raise the object the desired height. The gain in power will then be equal to the base divided by the height. By considering Fig. 126 ELEMENTS OF MECHANICS 137 FIG. 127. Power Applied Obliquely to Surface of Incline. further, it will be seen that in rolling the object up the incline the power will have to advance from the beginning of the incline to a point from which a line may be drawn per- ' pendicular to its di- rection to the top of the incline. In any case where the power is applied in any direction other than parallel with the incline, in roll- ing the object to the top, the power will have to advance to a point from which a line may be drawn perpendicularly to its direction to the top of the incline. In Figs. 127 and 128 are shown two other cases where the power is applied in a direction obliquely to the surface of the incline. In either of these cases, as in the other two cases, the gain in power will be found by dividing the distance through which the force distance through which the FIG. 128. Another Case where Power is Applied Obliquely to Surface of Incline. moves, ab, by the object is raised, cd. It will be further seen that the gain in power is 138 SELF-TAUGHT MECHANICAL DRAWING greatest when the direction in which the force is being applied is parallel with the incline. When the direction of the force is upward from the in- cline, as in Fig. 127, part of the force is expended in lifting the weight off from the incline, until, when its direction is made vertical, it is all expended in this way. When the direction of the force is downward from the incline, as in Figs. 126 and 128, part of it is lost in pressing the object against the incline. The Screw. The screw is a modified form of inclined plane, the lead of the screw, the distance FIG. 129. Differential Screw. that the thread advances in going around the screw once, being the height of the; incline, and the distance around the screw, measured on the thread, being the length of the incline. The Differential Screw. The differential screw is a compound screw having a coarse thread part of its length, and a somewhat finer thread the rest of its length, the object being to get a slow motion combined with the strength of a coarse thread. Fig. 129 shows such a screw. The piece A is a fixed part of some machine. The piston B slides within A, being prevented from turning by the pin C which enters a groove in B. If that part of the ELEMENTS OF MECHANICS 139 screw which engages in A has eight threads to the inch, and that part of it which engages in B has ten threads, then when the screw makes one revolution, it will advance into A one-eighth of an inch, and into B one- tenth of an inch ; the piston B will therefore advance through a distance equal to the difference between one-eighth of an inch and one-tenth of an inch, or twenty-five one-thou- sandths of an inch, requiring forty turns of the screw to make the piston advance one inch. Newton's Laws of Motion. The relation which exists between force and motion is stated by the three fundamental laws of motion formulated by Newton. Newton's first law says that if a body is at rest it will remain at rest, or, if it is in motion, it will continue to move at a uniform velocity in a straight line, until acted upon by some force and compelled to change its state of rest or of straight- line uniform motion. In a general way, this law is self-evident, and based on daily experience. How- ever, the part of the law stating that a body in motion will continue indefinitely to move if not acted upon by resisting forces, may not be so self- evident; yet whenever a body is brought to a stand- still after it has been in motion, such forces as frictional resistance, gravity, etc., always have in some way influenced the motion of the body. Newton's second law of motion says that a change in the motion of a body is proportional to the force causing the change, and takes place in the direction in which the force acts. If several forces act on a body, the change is proportional to 140 SELF-TAUGHT MECHANICAL DRAWING the resultant of the several forces, and takes place in the direction of the resultant. This has been clearly explained in the previous pages, in connec- tion with the resolution and composition of forces. The most important point to note in regard to the second law of motion is that when two or more forces act on a body at the same time, each causes a motion exactly the same as if it acted alone ; each force produces its effect independently, but the total effect on the motion of the body, of course, is a combination of all these independent motions. Newton's third law says that for every action there is an equal reaction. This means that if a force or weight presses downward on a support with a certain pressure, the reaction, or resistance in the support, must equal the same pressure. If a bullet is shot from a rifle with a certain force, there is a reaction, or "recoil," in the rifle, equal to the force required to give the velocity to the bullet. This law is very important, and many failures in machine design have been due to ignorance of the real meaning of the law of action and reaction. Newton's third law may be illustrated by a loco- motive drawing a train of cars. The driving wheels give as much of a backward push on the rails as there is of forward pull exerted on the train; and it is only because the rails are held in place by their fastenings, and by the weight rest- ing on them, that the locomotive is able to pull the train forward. This principle of action and reac- tion being equal and opposite is also an effectual ELEMENTS OF MECHANICS 141 bar to any perpetual-motion machine, as such a machine in order to work would have to produce a greater action in one direction than the reaction in the other direction. The Pendulum. A body or weight suspended from a fixed point by a string or rod, and free to oscillate back and forth is called a pendulum. The center of oscillation is the point which, if all of the material composing the pendulum, including the sustaining string or rod, were concentrated at it (the material so concentrated being considered as being suspended by a line of no weight) would vibrate in the same time as the actual pendulum. The length of the pendulum is the length from the point of suspension to the center of oscillation. When the length of the pendulum is unchanged, its time of vibration will be the same, if its angle of vibration does not exceed three or four degrees, and its time of vibration will be but slightly in- creased for larger angles. The time of vibration of a pendulum is not affected by the material of which it is made, whether light or heavy, except as the light mate- rial will offer greater resistance to the air, by presenting a greater surface in proportion to its weight, than a heavy material. The time of vibration of a pendulum of a given length is inversely as the square root of the inten- sity of gravity. As the intensity of gravity de- creases with the distance from the center of the earth it follows that a pendulum will vibrate faster at the poles or at sea level than it will at the equa- tor or at an elevation. 142 SELF-TAUGHT MECHANICAL DRAWING The time of vibration of a pendulum varies di- rectly as the square root of its length. That is, a pendulum to vibrate in one-half or one-third the time of a given pendulum will need to be only one- quarter or one-ninth of its length. Example l.A pendulum in the latitude of New York will require to be 39.1017 inches long to beat seconds. Required the length of a pendulum to make 100 beats per minute. A pendulum to make 100 beats per minute will have to make its vibrations in 60-100 of the time of one which is making 60 beats per minute, and its length will be equal to the length of one which beats seconds, multiplied by the square of 60-100, or: 39.1017 X 60 2 39.1017 X 3600 , ~W~ 10,000 : 14076 mches - Example #. Required the time of vibration of a pendulum 120 inches long. Letting x repre- sent the required time, we have the proportion V 120 :V 39. 1017 = x : 1, or 10.954 : 6.253 = x : 1. 10.954 , x = /? oco = 1-75 second. b.Zoo A short pendulum may be made to vibrate as slowly as desired by having a second "bob" placed above the point of suspension, which will partially counteract the weight of the lower bob. Falling Bodies. A falling body will have ac- quired a velocity at the end of the first second of 32.16 feet per second, under ordinary conditions. If the body is of such shape or material as to pre- sent a large surface to the air in proportion to its ELEMENTS OF MECHANICS 143 weight, its velocity will, of course, be lessened, and as its velocity depends upon the force of gravity, its velocity will be affected somewhat by the lati- tude of the place, and its distance above sea level. During the next second it will acquire 32.16 feet additional velocity, giving it a velocity of 64.32 feet at the end of the second second. Each suc- ceeding second will add 32.16 feet to the velocity the body had at the end of the preceding second. To find the velocity of a falling body at the end of any number of seconds, therefore, multiply the number of seconds during which the body has fallen by 32.16. This rule, expressed as a formula, would be : v = 32.16 X * in which v = velocity in feet per second, t = time in seconds. The acceleration due to gravity, 32.16 feet, is often, in formulas, designated by the letter g. As an example, find the velocity of a falling body at the end of the twelfth second: v = 32. 16X12 = 385. 92 feet. As the body falling starts from a state of rest, its average velocity will be one-half of its final ve- locity ; the distance through which it falls equals the average velocity multiplied by the number of seconds during which it has been falling. This rule, expressed as a formula, is : A-f X* .^ in which h = distance or height through which 144 SELF-TAUGHT MECHANICAL DRAWING body falls, and v and t have the significance given above. But v = 36.16 X t; if this value of v is in- serted in the formula just given, we have: This last formula, expressed in words, gives us the rule that the distance through which a body falls in a given time equals the square of the num- ber of seconds during which the body has fallen, multiplied by 16.08. How long a distance will a body fall in 10 sec- onds? Inserting t = 10 in the formula, we have: h = 16.08 f = 16.08 X 10 2 = 16.08 X 100 = 1608 feet. The time, in seconds, required for a body to fall a given distance equals the square root of the distance, expressed in feet, divided by 4.01. Ex- pressed as a formula, this rule would be : t = V ^ 4.01' As an example, assume that a stone falls through a distance of 3600 feet. How long time is required for this? Inserting h = 3600 in the formula, we have : V3600 60 t = . = = 15 seconds, very nearly. The velocity of a falling body after it has fallen through a given distance equals the square root of the distance through which it has fallen multi- plied by 8.02. ELEMENTS OF MECHANICS 145 This rule, expressed as a formula, is: What is the velocity of a falling body after it has fallen through a distance of 3600 feet? Inserting h = 3600 in the formula, we have: v = 8.02 X V3600 = 8.02 X 60 = 481.2 feet. The height from which a body must fall to acquire a given velocity equals the square of the velocity divided by 64.32. As a formula, this rule is: ~ 64.32 From what height must a body fall to acquire a velocity of 500 feet per second? Inserting v = 500 in the formula given, we have: 500 2 500 X 500 = 64.32 = .64.32 If a body is thrown upward with a given ve- locity, its velocity will diminish during each second at the same rate as it increases when the body falls. A body thrown up into the air in a vertical direction will return to the ground with exactly the same velocity as that with which it was thrown into the air. At any point, the velocity on the up- ward journey will be equal to the velocity on the downward journey, except that the direction is reversed. The acceleration of a falling body, 32.16 feet per second, is the value at the latitude of New York, at sea level. The force required to give to a falling body its 146 SELF-TAUGHT MECHANICAL DRAWING acceleration of 32.16 feet per second is the weight of the body itself. The force required to give any acceleration to a body, then, is to the weight of the body as that acceleration is to the acceleration produced by gravity. Therefore, to find the force required to produce a given rate of acceleration to a body, divide the weight of the body by 32.16, and multiply the quotient by the required rate of acceleration. Example. A body weighing 125 pounds is to be lifted with an acceleration of 10 feet per second. Required the strain on the sustaining rope. 125 00 ., X 10 = 38.8, the tension necessary to produce oZ.lb the acceleration. To this must be added the pull necessary to lift the weight without acceleration, or the weight of the body itself. Thus 38.8 + 125 = 163.8 is the re- quired tension on the rope. The rate of acceleration which a continuously acting force will produce is equal to the force divided by the weight of the body,, multiplied by 32.16. Energy and Work. The unit of work, the stand- ard by which work is measured, is the foot-pound, or the amount of work done in lifting a weight or overcoming a resistance of one pound through one foot of space. "Energy is the product of a force factor and a space factor. Energy per unit of time, or rate of doing work, is the product of a force factor and a velocity factor, since velocity is space per unit of time. Either factor may be changed at the ex- ELEMENTS OF MECHANICS 147 pense of the other; i.e., velocity may be changed, if accompanied by such a change of force that the energy per unit of time remains constant. Corre- spondingly force may be changed at the expense of velocity, energy per unit of time being constant. Example. A belt transmits 6000 foot-pounds per minute to a machine. The belt velocity is 120 feet per minute, and the force exerted is 50 pounds. Frictional resistance is neglected. A cutting tool in the machine does useful work ; its velocity is 20 feet per minute, and the resistance to cutting is 300 pounds. Then the energy received per minute - 120 X 50 = 6000 foot-pounds; and energy deliv- ered per minute = 20 X 300 = 6000 foot-pounds. The energy received therefore equals the energy delivered. But the velocity and force factors are quite different in the two cases." (Prof. A. W. Smith.) Force of the Blow of a Steam Hammer or Other Falling Weight The question, "With what force does a falling hammer strike ?" is often asked. This question can, however, not be answered directly. The energy of a falling body cannot be. expressed in pounds, simply, but must be expressed in foot-pounds. The energy equals the weight of the falling body multiplied by the distance through which it falls, or, expressed as a formula: E = WXh, in which E = energy in foot-pounds, W = weight of falling body in pounds, h = height from which body falls in feet. The energy can also be found by dividing the 148 SELF-TAUGHT MECHANICAL DRAWING weight of the falling body by 64.32 and then mul- tiplying the quotient by the square of the velocity at the end of the distance through which it falls. This rule, expressed as a formula, is: in which E and W denote the same quantities as before, and v = the velocity of the body at the end of its fall. Both of these formulas give, of course, the same results. That the second method gives the same result as multiplying the weight by the height through which it falls, is evident from the fact, stated under the head of "Falling Bodies/' that the square of. the velocity of a falling body, divided by 64.32, gives the height through which it has fallen. This second method allows of determining the energy of any weight or force moving at a given velocity, whether its velocity has been acquired by falling, or is due to other causes. Now assume that we wish to find the force of the blow of a 300-pound drop hammer, falling 2 feet before striking the forging, and compressing it 2 inches. The energy of the falling hammer when reach- ing the forging is: E = W X h = 300 X 2 = 600 foot-pounds. This energy is used during the act of compress- ing the forging 2 inches or 0.166 of a foot. Con- sequently, the average force of the hammer with ELEMENTS OF MECHANICS 149 which it compresses the forging is 600 -*- 0.166 + the weight of the hammer, or Average force of blow = A ,* + 300 = U.lbb 3600 + 300 = 3900 pounds. The general formula for the force of a blow is: in which F = average force of blow in pounds, W = weight of hammer in pounds, h = height of drop of hammer in feet, d = penetration of blow in feet. A horse-power, in mechanics, is the power ex- erted, or work done, in lifting a weight of 33,000 pounds one foot per minute, or 550 pounds one foot per second. The power exerted by a piston driven by steam or other medium during one stroke, in foot-pounds, is equal to the area of the piston, multiplied by the pressure per square inch, multi- plied by the stroke in feet, the product of the area by the pressure giving the force, and the stroke giving the distance through which the force is exerted. In the case of steam engines, where the steam is cut off at one-quarter, one- third or one- half of the stroke, the piston being driven the rest of the way by the expansion of the steam, the average pressure for the entire stroke, the ''mean effective pressure" (M.E. P.), as it is called, is the basis of calculations. As each revolution of the engine equals two strokes of the piston, the number of foot-pounds per minute an engine is developing will be the product of the area of the piston in 150 SELF-TAUGHT MECHANICAL DRAWING square inches, multiplied by the mean effective pressure, multiplied by the stroke in feet, multi- plied by the number of revolutions per minute times 2. This product, divided by 33,000, gives the indicated horse-power (I.H.P.) of the engine; this name being derived from the fact that the mean effective pressure is determined by the use of the steam engine indicator. Therefore: r TT D Area XM. E. P. X stroke X rev, permin. X2 LH ' P ' = 33,000 This formula may be transposed in various ways to give other information. For instance, if the piston area for a given horse-power is desired, then Area LH.P. X 33,000 M.E.P. X stroke X rev. per min. X 2. If the volume of the cylinder is desired, then A LH.P. X 33000 Area X stroke = , , ^ n rr^r M.E.P. X rev. permin. X 2. If the pressure to produce a given horse-power is desired, then MEp = LH.P. X 33000 Area X stroke X rev. per min. X 2. The mean effective pressure in the cylinder of the engine is, of course, considerably less than the boiler pressure as shown by the steam gauge. The indicated horse-power of an engine does not take into account the losses caused by the friction of the working parts. The power which the engine actually delivers as shown by a brake dynamo- meter or other contrivance at the flywheel is called the brake horse-power. CHAPTER IX FIRST PRINCIPLES OF STRENGTH OF MATERIALS Factor of Safety. It is obvious that it would be unsafe in designing a piece of construction work to allow a strain of anywhere near the breaking limit of the material it is to be made from. It is, therefore, customary in making any calculations for the size of the parts to use what is called a factor of safety, by making the part from three or four to ten or even more times the strength neces- sary to just resist breaking with a steady load. The factor of safety used will depend upon several considerations. It will depend, first, upon the na- ture of the material used. A wrought or drawn metal, for instance, will be likely to be more uni- form in its nature than a cast metal which may contain air holes, or which may be more or less spongy, or which may be under unequal strains in cooling. The matter of strains in a casting due to unequal cooling is to a considerable extent a mat- ter of proper or improper design ; still it is not possible to entirely avoid them. Again the factor of safety to be used will depend upon the nature of the work which will be re- quired of the part. If the part has to simply sus- tain a steady load it will not need to be as strong as though the load was applied and reversed, or 151 152 SELF-TAUGHT MECHANICAL DRAWING even as strong as though the load was applied and released. To illustrate, it is a familiar fact that a piece of wire which may be bent a given amount without apparent injury, may be broken by repeat- edly bending it back and forth the same amount at one point. And, similarly, in machine parts, rupture may be caused not only by a steady load which exceeds the carrying strength, but by re- peated applications of stresses none of which are equal to the carrying strength. Rupture may also be caused by a succession of shocks or impacts, none of which alone would be sufficient to cause it. Iron axles, the piston rods of steam hammers and other pieces of metal subjected to repeated shocks, invariably break after a certain length of service. The factor of safety used will therefore vary widely with the nature of the work required of the part. For a steady or "dead" load, Prof. A. W. Smith says: "In exceptional cases where the stresses permit of accurate calculation, and the material is of proven high grade and positively known strength, the factor of safety has been given as low a value as 1J but values of 2 and 3 are ordinarily used for iron or steel free from welds ; while 4 to 5 are as small as should be used for cast iron on account of the uncertainty of its composi- tion, the danger of sponginess of structure, and indeterminate shrinkage stresses." Others would make 3 the lowest factor of safety that should be used for wrought iron and steel. Where the load is variable, but well within the elastic limit of the material, that is where the load is not so great but so that the part will immedi- STRENGTH OF MATERIALS 153 ately resume its original shape when the load is removed, a factor of safety of 5 or 6 might be used. The part will need to be made stronger if the load or force acts first in one direction and then in the opposite direction, that is, if it acts back and forth, than it will need to be if the same force is simply applied and then released. Where the part is subjected to shock, the factor of safety is generally made not less than 10. A factor of safety as high as 40 has been used for shafts in mill-work which transmit very variable powers. In cases where the forces are of such a nature that they cannot be determined, then Prof. Smith says: "Appeal must be made to the precedent of successful practice, or to the judgment of some ex- perienced man until one's own judgment becomes trustworthy by experience. * * * In proportioning machine parts, the designer must always be sure that the stress which is the basis of calculation or the estimate, is the maximum possible stress; otherwise the part will be incorrectly propor- tioned." And he cites the case of a pulley where if the arms were to be designed only to resist the belt tension they would be absurdly small, because the stresses resulting from the shrinkage of the casting in cooling are often far greater than those due to the belt pull. In many cases the practical question of feasi- bility of casting will determine the thickness of parts, independent of the question of strength. For instance, on small brass work, such as plumb- ers' supply, and small valve work, a thickness of about 3-32 of an inch is as little as can be relied 154 SELF-TAUGHT MECHANICAL DRAWING on to make a good casting on cored out work ; or in the case of partitions in such work where the metal has to flow in between cores, a thickness of about J of an inch is as small as should be used; yet such thicknesses may be much greater than are required to give the necessary strength. On larger cast iron work, the thickness to be allowed to insure a good casting will, of course, depend upon the size of the piece. The judgment of the pattern-maker or foundry-man will naturally de- termine the thickness in such cases. Shape of Machine Parts. While the size of ma- chine parts will vary greatly with the nature of the work required of them, their shape will depend very much on the manner or direction in which the load or strain is brought to bear upon them. If the part is subjected to simple tension, that is, merely resists a force tending to pull it apart, then the shape of the member which serves this purpose is not very material, though a round rod, being most compact and cheapest, is best. Almost any shape will answer, however, though it is well to avoid using thin and broad parts, as a strain, though not greater than that which the part as a whole might bear safely, might be brought upon one edge, pro- ducing^a tearing effect beyond the safe limit. For resisting simple tension the part should be made of uniform size its entire length, of a size to be deter- mined by the tensile strength of the material and the factor of safety used. If the part is to resist compression, then when the proportion of its length to its diameter or thickness is such that it will " buckle' ' or bend, STRENGTH OF MATERIALS 155 instead of crushing, that is when its length ex- ceeds five or six times its diameter, it becomes desirable to use a hollow or cross-ribbed form of construction, so as to get the metal as far from the axis of the piece as possible. The hollow cylind- rical form, by getting all of the metal equally dis- tant from the axis is, of course, most effective, but considerations of appearance may make a hol- low square form more desirable, while considera- tions of cost may make a cross-ribbed form to be preferred, as such a form can be cast without the use of cores. In cases where a wrought metal must be used a solid form is often the only practicable one. When it becomes important to keep the weight down to the lowest point, it is common to have the piece slightly enlarged in the middle of its length, as in the case of connecting rods of steam engines. In the case of steam engine con- necting-rods, the tendency to buckle is least side- ways, as the cross-head and crank-pins tend to hold it in line this way, while the rotary motion of the crank-pin tends to produce buckling the other way. Connecting rods are therefore frequently made somewhat flat, of a breadth about twice their thickness. When a piece is designed to resist bending, it becomes desirable to get a good depth of material in the direction in which the force is applied, as the capacity of a piece to resist bending increases as the square of its thickness or depth in the di- rection of the force, but only directly as its breadth or width, so that to increase the thickness of a piece two or three times in the direction of the 156 SELF-TAUGHT MECHANICAL DRAWING force would increase its capacity to resist bending four or nine times; while to increase its breadth two or three times would only increase its strength two or three times. The proportion of depth to breadth which can be used will, of course, depend upon the length of the piece, as if the piece is long and its depth is made large in proportion to its thickness the tendency will be for the piece to buckle, or yield sideways. To resist this tendency it is customary to put ribs on the edges of such a FIG. 130. FIG. 131. FIGS. 130 and 131. Beam Cross-sections of Different Types. piece, giving it the form shown in Fig. 130. The hollow box-form shown in Fig. 131 is, of course, equally effective to resist combined bending and buckling stresses, and in some cases may be pref- erable as a matter of appearance on account of the impression of solidity which it gives. A projecting beam, like that shown in Fig. 132, designed to resist a force or sustain a load at its end, would need to have its lower edge made of the form of a parabola, if made of uniform thickness. If the edges were ribbed to prevent buckling, then material might be taken out of the middle portion, as shown in Fig. 133, without weakening it. STRENGTH OF MATERIALS 157 Strength of Materials as Given by Kirkaldy's Tests. A very large number of tests of cast iron made by Kirkaldy gave results as follows : Tensile strength per square inch, necessary to just tear asunder, from about 10,000 or 12,000 pounds to about 28,000 or 32,000 pounds, or an average strength of about 20,000 pounds. Tests on the ability of cast iron to resist crushing gave results vary ing from about 50, 000 to about 150, 000 pounds, FIG. 132. Cantilever of Uniform Strength, when Loaded at End. FIG. 133. Common Design of Cantilever of Uniform Strength. or an average strength of about 100,000 pounds per square inch. These tests indicate that cast iron has about five times the capacity to resist crushing that it has to resist tension. They also indicate that cast iron is a somewhat uncertain material. Tests of wrought iron indicated a tensile strength of between 40,000 and 50,000 pounds per square inch, the elastic limit being reached at about one- half the tensile strength. Tests on steel castings gave results for tensile strength ranging from 55,000 to about 64,000 pounds per square inch, 158 SELF-TAUGHT MECHANICAL DRAWING the elastic limit being reached at about 30,000 pounds. Tests of wire gave results as follows : Brass, from 81,000 to 98,000 pounds per square inch of area. Iron, from 59,000 to 97,000 pounds. Steel, from 103,000 to 318,000 pounds. The tensile strength of regular machine steel (low carbon steel) is generally given at about 60,000 pounds per square inch. Size of Parts to Resist Stresses. To resist ten- sion it is, of course, only necessary to have the piece of such a size that each square inch shall not have a stress greater than the average strength of the material (as 20,000 pounds for cast iron) divided by whatever factor of safety may be selected. To Resist Crushing. Prof. Hodgkinson's rule for the strength of hollow cast iron pillars is as follows : To ascertain the crushing weight in tons multiply the outside diameter by 3.55; from this subtract the product of the inside diameter multi- plied by 3.55, and divide by the length multiplied by 1.7. Multiply this quotient by "46.65. Ex- pressed as a formula this rule would be: (D X 3.55) - (d X 3. 55) L X 1.7 S c = 46.65 X in which Sc = ultimate compressive (crushing) strength of hollow column, in tons, D = outside diameter in inches, d = inside diameter in inches, L = length of column in feet. STRENGTH OF MATERIALS 159 Any desired factor of safety may be introduced in the above formula by dividing the factor 46.65 by the factor of safety. In this case the formula would be: Q 46J>5_Xl CD ._X_SL55) - (d X 3.55)] F X L X 1.7 in which S = safe compressive strength in tons, F = factor of safety, and D, d and L have the same meaning as above. This rule and formula assumes that the ends of the column are perfectly flat and square, and that the load bears evenly on the whole surface. If the ends are rounded, the column yields at about one-half the -stress of one with fixed square ends. To Resist Bending. In the following commonly given rules for the strength of beams or bars to -i K-H I FIG. 134. Rectangular Cantilever. resist breaking by transverse stresses, the tensile strength of cast iron is assumed at 20,000 pounds per square inch. Divide 20,000 in the formulas > 160 SELF-TAUGHT MECHANICAL DRAWING by the desired factor of safety. The breadth and depth of rectangular bars, the diameter, if the bar is round, and the length, are all in inches. For rectangular bars fixed at one end with the force applied at the other, Fig. 134, the breaking load equals bX d 2 X 20,000 I : For round bars under the same conditions, Fig. 135, the breaking load equals JL / 0.59 X d*X 20,000 6 I If the rectangular bar is hollow, as shown in FIG. 135. Circular Section Cantilever. Fig. 131, subtract the internal b X d 2 from the external b X d 2 . If the round bar is hollow subtract the internal d 3 from the external d 3 . The case of a bar of the I-section shown in Fig. 130 is similar to that of the hollow rectangular bar of Fig. 131, the depressions in its sides correspond- ing to the hollow part of Fig. 131, the sum of their STRENGTH OF MATERIALS 161 depths corresponding with the internal width b of the hollow rectangular bar. If a beam is fixed at one end and the load is evenly distributed throughout its entire length, it will bear double the weight it will if the load is supported at the outer end. If the beam is supported at the ends and loaded in the middle it will bear four times the weight of the beam of Fig. 134, or, if the load is evenly dis- tributed throughout the length of the beam, eight times. If the beam, instead of being simply supported at- the ends, has the ends fixed and is loaded at the center, its ability to resist breaking will be doubled as compared with that when loaded at the center and with the ends only supported. Regarding the safe load that beams or bars of different material may bear Griffin says that "with but a general knowledge of the elastic limit, ordi- nary steel is good for from between 12,000 to 15,000 pounds per square inch non-reversing stress, and from 8000 to 10,000 pounds reversing stress. Cast iron is such an uncertain metal, on account of its variable structure, that stresses are always kept low, say from 3000 to 4000 for non-reversing stress, and 1500 to 2500 for reversing stress. " Again, though the tests of wrought iron show it to have a much higher tensile strength than cast iron, Nystrom, in formulas for lateral strength, gives wrought iron but little more than three-quar- ters the value of cast iron, probably because it bends so readily. 162 SELF-TAUGHT MECHANICAL DRAWING A table is appended giving the average breaking strength, in pounds per square inch, of some com- monly used materials in engineering practice. Tension. Compression. Aluminum 15,000 12,000 24,000 30,000 Copper cast 24 000 40,000 Iron cast 15,000 80,000 Iron wrought .... 48,000 46,000 Steel castings 70,000 70,000 Structural steel .... 60,000 60,000 Stresses in Castings. Reference has been pre- viously made to stresses in castings, due to shrink- age in cooling. If all parts of a casting could be made to cool equally fast there would not be much trouble in this respect, but as different parts of a casting vary in thickness, the time they require to cool will vary, and the thick parts remaining fluid the longest, will, on cooling, cause a strain on the already cool thin parts. In the case of a pulley, where the rim and arms are much lighter than the hub, the hub on cooling will tend to draw the arms to itself and away from the rim, and if the differ- ence in thickness is great, they may be even found to be pulled away so as to show a crack where they join the rim. The"remedy in such a case would, of course, be first, to take out as much of the metal from the center of the hub as possible by means of a core, and second, to keep the outside of the hub as small as would be consistent with strength, getting necessary thickness for set screws by hav- ing a raised place or boss at that point. STRENGTH OF MATERIALS 163 As these strains are primarily due to unequal cooling, it is evident that in order to reduce them to the lowest point the first thing to do is to make the different parts of the casting of as nearly uni- form thickness as possible. Where different parts of the casting vary in thickness, the change from one thickness to the other should be made as grad- ual as possible. Sharp internal corners should also be avoided, as such places are very liable to be spongy; the sand from the sharp corner in the mould is also very liable to wash away when the metal is poured in, and lodge in some other place, causing a defective casting. A good ' ' fillet, " as an internal round corner is called, which the pattern- maker may put into the pattern with wax, putty or leather, will not be very expensive, and will save much trouble in the casting. Besides possessing a knowledge of factors of safety, proportioning parts to resist various stresses and the like, a general knowledge of the principles of foundry and machine shop practice is essential to properly design machine work. If one does not understand foundry work, he will be con- stantly designing castings which it will be im- practicable to mould ; if not actually impossible of moulding, they will be needlessly expensive. And in like manner, unless he understands the general principles of machine shop practice, his work will be giving trouble at that end of the line. CHAPTER X CAMS General Principles. In designing machinery it is frequently desirable to give to some part of the mechanism an irregular motion. This is often done by the use of cams, which are made of such form that when they receive motion, either rotary or reciprocating, they impart to a follower the desired irregular motion. The follower is sometimes flat, and sometimes round. When the follower is round it is usually made in the form of a wheel or roller, so as to les- sen the wear and the friction. The follower may work upon the edge of the cam, or if round, it may work in a groove formed either on the face or on the side of the cam. The working surfaces of cams with round fol- lowers are laid out from a pitch line, so called, which passes through the center of the follower. The shape of this pitch line determines the work which the cam will do. The working surface of the cam is at a distance from the follower equal to one-half the diameter of the follower. This prin- ciple of a pitch line holds good whether the cam works only upon its edge like the one shown in Fig. 139, or whether it has an outer portion to insure the positive return of the follower. This 164 CAMS 165 outer portion is frequently made in the form of a rim of uniform thickness around the groove. Design a Cam Having a Straight Follower Which Moves Toward or From the Axis of the Cam, as Shown in Fig. 136. Let it be required that the follower shall advance at a uniform rate from a to FIG. 136. Cam with Straight Follower having Uniform Motion. 6 as the cam makes a half revolution, this advance being preceded and followed by a period of rest of a twelfth of a revolution of the cam. Divide that half of the cam during the revolu- tion of which the follower is to be raised from a to &, in this case the half at the right of the vertical center line, into a number of equal angles, and 166 SELF-TAUGHT MECHANICAL DRAWING divide the distance from a to b into the same num- ber of equal spaces. Mark off the points so ob- tained onto the successive radial lines as indicated by the dotted lines, and at the points where these dotted lines intersect the radial lines draw lines at right angles to the radial lines to represent the position of the follower when these radial lines become vertical as the cam revolves. A period of rest in a cam is represented by a cir- cular portion, having the axis of the cam as its center. In order, therefore, to obtain the required periods of rest, the distances of a and b from the center are marked off upon the radial lines c and d, these lines being made a twelfth of a revolution from the vertical center line, and lines represent- ing the follower are drawn at these points as be- fore. To get the return of the follower the space from c to d is divided into a number of equal angles, and the distance from e to /is divided off to represent the desired rate of return of the follower. In this case the rate of return is made uniform, so the distance ef is spaced off equally. The distance of these points from the axis is marked off upon the radial lines between c and d, and lines representing the follower are drawn. A curved line, which may be made with the aid of the irregular curves, which is tangent to all of the lines representing the follower, gives the shape of the cam. Fig. 137 shows a cam having the conditions as to the rise, rest and return of the follower the same as the one shown in Fig. 136, the follower, how- ever, being pivoted at one end. CAMS 167 Draw the arc ab representing the path of a point in the follower at the vertical center line, and divide that part of the arc through which the fol- lower rises into the same number of equal spaces as the half circle at the right of the vertical cen- ter line is divided into angles. Through these FlG. 137. Cam with Pivoted Follower. points draw lines, as shown, representing consecu- tive positions of the working face of the follower. The various distances of the follower from the axis of the cam are now marked off upon the corre- sponding radial lines as before. Lines to represent the follower are now drawn across each of these radial lines, at the same angle to them that the follower makes with the vertical center line when 168 SELF-TAUGHT MECHANICAL DRAWING at that part of its stroke corresponding to the par- ticular radial line across which the line represent- ing the follower is being drawn. A curved line passing along tangent to all of these lines gives the shape of the cam as before. Design a Cam with a Round Follower Rising Ver- tically. In Fig. 138 the follower has the same uni- form rise, and the same periods of rest as before. FIG. 138. Cam with Roller Follower. A cam with a round follower is less limited in its capabilities than one with a straight follower ; in the one here shown the follower on its return drops below the position in which it is shown. That part of the cam during which the conditions are the same as in the others is divided off and CAMS 169 the position of the center of the follower upon the radial lines is obtained in the same manner as before. That part of the cam representing the return of the follower is divided into such angles as desired, and the distance through which the fol- lower is to drop as the cam revolves through each of these angles is marked off upon the proper radial line. A curved line which is now made to pass through all of the points so obtained gives the pitch line of the cam. In drawing such a cam it is not always neces- sary to fully draw the working faces. The pitch line and the method of obtaining it being shown, a number of circles representing consecutive posi- tions of the follower may be drawn. This will usually be sufficient. The side view of the cam, which in a case like this would naturally be made in section, will give opportunity to show any fur- ther detail that may be desired. Design a Cam with a Round Follower Mounted on a Swinging Arm. Fig. 139 shows such a cam, all of the conditions as to rise, rest and return of the follower being the same as in the cam shown in Fig. 138. The cam is divided into the same angles as before, and the position of the follower is laid out on these radial lines as though it moved ver- tically. These positions are then modified in the following manner : Draw the arc ab representing the path of the center of the follower as it rises, and extend the dotted circular lines, which repre- sent successive heights of the follower, from the vertical center line to this arc. The distance of each of the intersections of the dotted circular 170 ELF-TAUGHT MECHANICAL DRAWING lines with the arc a&, from the vertical center line is then taken with the compasses and is marked off upon the same dotted line from the radial line at which it terminates, or, where the follower has a period of rest, from both of the radial lines FIG. 139. Cam with Roller Follower Mounted on Swinging Arm. where the period of rest takes place. Thus the dis- tance of the point 1 from the vertical center line is marked back upon the dotted circular line from the radial lines ra and n. Point 2 is marked back from the radial line o. Point 3 is marked back from the line p. By this means the position which the fol- lower will occupy, when each of the radial lines has become vertical, as the cam revolves, is deter- CAMS 171 mined. A curved line which is made to pass through all of these points will be the required pitch line of the cam. The method of getting the working face of the cam is indicated by the small dotted circular arcs, which are drawn with a radius equal to that of the follower. It will be noticed that, as the follower, on its return, drops below the position in which it is shown, it passes to the other side of the vertical center line, so that in marking off its position from the radial lines x and y this must be borne in mind. The question as to FIG. 140. Reciprocating Motion Cam. on which side of a radial line the new position of the follower will be, may be readily determined by imagining the cam to revolve so as to bring that particular line vertical. Reciprocating Cams. Fig. 140 shows a straight cam, which by a reciprocating motion imparts a sideways motion to its follower. The pitch line of such a cam may be determined by intersecting lines at right angles to each other. As here shown the distance through which the follower is to be raised is divided into a number of equal spaces by horizontal lines, and the distance through which it is desired to have the cam move in order to raise the follower from one horizontal line to the next one is indicated by vertical lines. A curved line 172 SELF-TAUGHT MECHANICAL DRAWING which is made to pass through the intersections of these lines will be the required pitch line of the cam. If the follower, instead of rising vertically, rose at ah angle, or if it were mounted on a swinging arm, the pitch line would be modified in the same manner as that of the cam shown in Fig. 139. Cams With a Grooved Edge. It is sometimes de- sired to have a revolving cam impart a sideways FIG. 141. Cam with Grooved Edge. motion to a follower. This is done by having a groove in the edge of the cam, as shown in Fig. 141. Such a cam may be considered as a modified form of a reciprocating cam, and its pitch line may be determined in the same way. By laying out a development of the pitch lino, or of that part of it which is to operate the follower, as shown in Fig. 142, horizontal lines, that is, lines parallel with the pitch line, may be drawn to indi- cate successive stages in the movement of the fol- lower, and lines at right angles to these to indicate CAMS 173 the desired movement of the cam. The pitch line is then drawn through the intersections of these lines as before. A Double Cam Providing Positive Return. In a cam like that shown in Fig. 138, where the return FIG. 142. Development of Cam Action of Grooved-Edge Cam in Fig. 141. of the follower is insured by a groove in the face of the cam, the groove must be slightly broader than the diameter of the cam roller to insure free- dom of action, as, when the cam is forcing the rol- FIG. 143. Double Cam Providing Positive Return. ler away from the center, the roller will revolve in the opposite direction to that in which it revolves when the other face of the cam groove acts on it to draw it toward the center, so that unless clear- 174 SELF-TAUGHT MECHANICAL DRAWING ance is provided, there will be a grinding action between the roller and the faces of the cam groove. This clearance, however, causes the cam to give a knock or blow on the roller each time its action is reversed, and the reversal of the direction of the revolution of the roller itself causes a temporary grinding action. These actions may become ob- FlG. 144. Positive Return Cam with Rollers Mounted on Swinging Arms. jectionable, especially at high speeds. A method which overcomes these objections, and which is preferred by some for such work, is shown in Fig. 143, where the return is secured by a secondary cam mounted on the same shaft as the primary cam, but acting on a roller of its own. In this case there is no reversal of the direction of the revolu- tion of the rollers, so that the necessity of provid- CAMS 175 ing clearance does not exist. Where the forward and backward motion of the rollers is in a straight line passing through the center of the cam shaft, as in this case, it is only necessary in designing the secondary cam to preserve the distance be- tween its pitch line and the pitch line of the prim- ary cam constant, measuring through the center of the cam shaft, as shown at x and y. If, however, the rollers are mounted on swing- ing arms, as shown in Fig. 144, so that their for- ward and backward motion is not in such a straight line, then the shape of the secondary cam will be subject to modification on principles previously explained. It is obviously necessary where this method of operation is used, that provision be made to absolutely prevent any change in the relative position of the two cams, as by bolting them to- gether, or, better still, by having them cast together in one piece. Cams for High Velocities. In machinery work- ing at a high rate of speed, it becomes very im- portant that cams are so constructed that sudden shocks are avoided when the direction of motion of the follower is reversed. While at first thought it would seem as if the uniform motion cam would be the one best suited to conditions of this kind, a little consideration will show that a cam best suited for high speeds is one where the speed at first is slow, then accelerated at a uniform rate until the maximum speed is reached, and then again uni- formly retarded until the rate of "motion of the follower is zero or nearly zero, when the reversal takes place. A cam constructed along these lines 176 SELF-TAUGHT MECHANICAL DRAWING FIG. 145. Uniformly Accelerated Motion Cam. CAMS 177 is called a uniformly accelerated motion cam. The distances which the follower passes through during equal periods of time increase uniformly, so that, if, for instance, the follower moves a distance equal to 1 length unit during the first second, and 3 during the second, it will move 5 length units during the third second, 7 during the fourth, and so forth. When the motion is retarded, it will move 7, 5, 3 and 1 length units during successive seconds, until its motion becomes zero at the re- versal of the direction of motion of the follower. In Fig. 145 is shown a uniformly accelerated motion plate cam. Only one-half of the cam has been shown complete, the other half being an exact duplicate of the half shown, and constructed in the same manner. The motion of the follower is back and forth from A to G, the rise of the cam being 180 degrees, or one-half of a complete revolution. To construct this cam, divide the half-circle, AKL, in six equal angles, and draw radii HB , HC^ , etc. Then divide AG first in two equal parts AD and DG, and then each of these parts in three divisions, the length of which are to each other as 1:3:5, as shown. Then with H as a center draw circular arcs from J5, C, Z>, etc., to B , d , A , etc. The points of intersection between the circles and the radii are points on the cam surface. If the half-circle AKL had been divided into 8 equal parts, instead of 6, then the line AG would have been divided into 8 parts, in the proportions 1:3:5:7:7:5:3:1, each division being the same amount in excess of the previous division while the motion is accelerated, and the same amount 178 SELF-TAUGHT MECHANICAL DRAWING less than the previous division while the motion is being retarded. With a cam constructed on this principle the follower starts at A from a velocity of zero ; it reaches its maximum velocity at D ; and at G the velocity is again zero, just at the moment when the motion is reversed. A graphical illustration of the shape of the uni- formly accelerated motion curve is given in Fig. i BC DE F GHL \ / FIG. 146. Development and Projection of Uniformly Accelerated Motion Cam Curve. 146. To the right is shown the development of the curve as scribed on the surface of a cylindrical cam. This development is necessary for finding the projection on the cylindrical surface, as shown at the left. To construct the curve, divide first the base circle of the cylinder in a number of equal CAMS 179 parts, say 12; set off these parts along line AL, as shown ; only one division more than one-half of the development has been shown, as the other half is the same as the first half, except that the curve to be constructed here is falling instead of rising. Now divide line AK in the same number of divisions as the half-circle, the'divisions being in the proportion 1:3:5:5:3:1. Draw horizontal lines from the divisions on AK and vertical lines from B, C, D, etc. The intersections between the two sets of lines are points on the developed cam curve. These points are transferred to the cylindrical surface at the left simply by being projected in the usual manner. In order to show the difference between the uni- form motion cam curve, and that illustrating the uniformly accelerated motion, a uniform motion cylinder cam has been laid out in Fig. 147. The base circle is here divided in the same number of equal parts as the base circle in Fig. 146. The divisions are set off on line AL in the same way. The line AK, however, is divided into a number of equal parts, the number of its divisions being the same as the number of divisions in the half -circle. By drawing horizontal lines through the division points on AK, and vertical lines through points B, C, D, etc., points on the uniform motion cam curve are found. It will be seen that this curve is merely a straight line AM. The curve is transferred to its projection on the cylinder surface at the left, as shown. It is evident from the developments of the two curves in Figs. 146 and 147, that the uniform motion 180 SELF-TAUGHT MECHANICAL DRAWING curve, Fig. 147, causes the follower to start very abruptly, and to reverse from full speed in one direction to full speed in the opposite direction. The uniformly accelerated motion curve, Fig. 146, permits the follower to start and reverse very smoothly, as is clearly shown by the graphical A B C DEFGHIL FIG. 147. Development and Projection of Uniform Motion Cam Curve. illustration of the curve. The abrupt starting and reversal of the follower in the uniform motion curve is the cause why this form of cam, while the simplest of all cams to lay out and cut, cannot be used where the speed is considerable, without a perceptible shock at both the beginning and the end of the stroke. CAMS 181 Besides the uniformly accelerated motion cam curve, quite commonly called the gravity curve, on account of it being based on the same law of acceleration as that due to gravity, there is another curve, the harmonic or crank curve, which is quite often used in cam construction. The harmonic motion curve provides for a gradual increase of speed at the beginning, and decrease of speed at the end, of the stroke, and in this respect resembles xV \ EI. FI GI H! FIG. 148. Lay-out of Harmonic Motion Cam Curve. the uniformly accelerated motion curve; but the acceleration, not being uniform, does not produce so easy working a cam as the gravity curve provides for. The harmonic motion curve is, however, very simple to lay out, and for ordinary purposes, where excessively high speeds are not required of the mechanism, cams laid out according to this curve are very satisfactory. The harmonic curve is laid out as shown in Fig. 148. Draw first a half-circle AEL Divide the 182 SELF-TAUGHT MECHANICAL DRAWING circle in a certain number of equal parts. Draw a line AI /! , and divide this line in a number of equal parts, the number of divisions of A v J t being the same as that of the half -circle. Now draw hori- zontal lines from the divisions A, B, C, etc., on the half-circle, and vertical lines from the divisions on line AI /! . The points where the lines from corre- sponding division points intersect, are points on the required harmonic cam curve. An approximation of the uniformly accelerated motion or gravity curve can be drawn as shown in \ \ n \ \ / ^ y r ^^ ^^ A i Bj c, D, e i F., G! Hj i, FIG. 149. Approximation of Uniformly Accelerated Motion Curve. Fig. 149. By using this approximate method, any degree of accuracy can be attained without the necessity of dividing the vertical line AK, Fig. 146, in an excessively great number of parts. The approximate curve in Fig. 149 is constructed as follows: Draw a half-ellipse AEI, in which the minor axis is to the major axis as 8 to 11. Divide this half-ellipse in any number of equal parts, and divide the line Ail t in the same number of equal parts. Now draw horizontal lines from the division CAMS 183 points on the ellipse, and vertical lines BI, Ci, etc. The points of intersection between corresponding horizontal and vertical lines, are points on the cam curve. This cam curve, as well as the one in Fig. 148, can be transferred to the cylindrical surface of a cylinder cam by ordinary projection methods, as shown in Figs. 146 and 147. In Figs. 150 and 151 are shown two plate cams for comparison. The one in Fig. 150 is a uniform V FIG. 150. Plate Cam Laid Fia. 151. Plate Cam Laid out for Uniform Motion. out for Uniformly Accel- erated Motion. motion cam. The dwell is 180 degrees, the rise, 90 degrees, and the fall, 90 degrees. As shown by the sudden change of direction of the cam curve at A and B, there is considerable shock when the follower passes from its "dwell" to the " rise, " as well as at the end of the ' ' fall. ' ' A sudden reversal takes place at C, which also causes a shock in the mechanism connected with the follower. In the uniformly accelerated motion cam, Fig. 151, the 184 SELF-TAUGHT MECHANICAL DRAWING passing from " dwell" to "rise, " the reversal of the direction of motion, and the return to the "dwell" position, is accomplished by means of smoothly acting curves, and, even at high speeds, no per- ceptible shock will be noticed. The examples given will show the necessity of careful analysis of conditions, before a certain type of cam curve is selected. In machinery which works at a low rate of speed, it is not important whether the follower moves with a uniform, har- monic, or uniformly accelerated motion ; but when the cam has a high rotative speed, and the follower a reciprocating motion, it often becomes practically impossible to make use of the uniform motion curve in the cam. In such cases, as already men- tioned, the harmonic, or, preferably, the uniformly accelerated motion curve should be used in laying out the cam. CHAPTER XI SPROCKET WHEELS WHEN it is desired to transmit power from one shaft to another one quite near to it, especially if the power to be transmitted is considerable, so as to preclude the use of belting, sprocket wheels with chain are frequently used, if the speed is not high. Bicycles afford a familiar illustration of this sort of power transmission. Fig. 152 shows a sprocket wheel of a type similar to those used on bicycles and shows the method of getting the shape of the teeth. The chain is shown with the links (on the side toward the observer) removed so as to allow of showing the teeth with- out dotted lines. The size of a sprocket wheel to fit a given chain may be determined graphically as follows : A circle, not shown in the illustration, is first drawn of a diameter about equal to that of the desired wheel, and this circle is spaced off into as many divisions as the wheel is to have teeth. Lines corresponding to the dotted radial lines in the upper half of the wheel shown, are drawn from these division points to the center of the circle. A templet, similar in shape to that shown in Fig. 154, is next cut out of paper, the lines ab and cd being at right angles to each other, and the length of a link of the chain, measured from center to center 185 186 SELF-TAUGHT MECHANICAL DRAWING of the pins as shown at a, Fig. 152, is marked off upon the line ab, measuring equally each way from the center line cd. In getting the length of the link in the chain it will be best, for the sake of ac- curacy, to measure off the length of a considerable portion of the chain, and with the spacing com- passes divide this length into twice as many spaces as there are links in the measured portion of the FIG. 152. Sprocket Wheel and Chain. chain. The compasses, being then set to exactly half the length of a link, may be used to mark off the length of the link, 1 2, upon the templet. Now letting the angle abc, Fig. 155, represent one of the angles into which the circle has been di- vided, bisect it to get a center line bd, and placing the templet so that its line cd shall coincide with this center line move it along until the points 1 2 shall coincide with the lines ab and cb of the angle. These points being now marked off upon the lines, give the location of the centers of the pins in the chain, and a line connecting them will be one side SPROCKET WHEELS 187 of the polygon which forms the pitch line of the wheel. A spiral may now be formed upon this polygon (see geometrical problem 19, Figs. 41 and 42) , and will give the path of the pin as the chain FIG. 153. Sprocket Wheel Designed for Common Link Chain. unwinds from the wheel when the latter revolves, as shown in Fig. 152. The working face of that part of the tooth in the wheel lying outside of the pitch polygon is now struck from such a center as will cause it to fall slightly within the path of the chain, as just obtained, so that the link may fall FIG. 154. FIG. 155. FIGS. 154 and 155. Graphical Method of Laying Out Sprocket Wheel. freely into place as it enters upon the tooth. Of course allowance must be made all around for the natural roughness of the casting if the wheel is to be left unfinished. The length of the tooth is usually made about equal to the width of the chain. 188 SELF-TAUGHT MECHANICAL DRAWING If a wheel is to have many teeth, it will gener- ally be accurate enough to consider the pitch line as a circle of a circumference equal to the number of the teeth multiplied by the length of the link. Its diameter will then, of course, be found by dividing the circumference by 3.1416. In the case of the wheel shown in Fig. 152, should the pitch line be regarded as such a circle it would have a diameter a little over a thirty- second of an inch too small, if the length of the link is taken at three-quarters of an inch. If the wheel were to be made twice as large, the error would be a little less than a sixty-fourth of an inch, as it would decrease at a slightly faster rate than that at which the number of the teeth increased. An error of a sixty-fourth of an inch in the diameter of such a sprocket would be of but very little moment. Where a sprocket has but few teeth, however, it will be on the side of safety to always give to the pitch line its true polygonal form, and the only way by which its diameter could be ascer- tained with any greater accuracy than by the method here given would be to calculate it, as may be done by trigonometry. When the pitch line of a sprocket is regarded as a circle, the path of the chain as it unwinds will be regarded as an involute (see geometrical problem 20). The shape of the rim of a sprocket wheel will be governed by the style of the chain for which it is designed. Fig. 153 shows a portion of the rim of a wheel which is designed for a common link chain ; but whatever the general shape of the rim may be, the working faces of the teeth, or of the SPROCKET WHEELS 189 projections. which correspond to teeth, will always be made on the principles here explained. The speed ratio of the two wheels of a pair of sprockets will be inversely as the number of teeth in each. For instance, if the large and the small wheels have respectively 13 and 7 teeth, then the speed of the large wheel will be to the speed of the small wheel as 7 to 13. CHAPTER XII GENERAL PRINCIPLES OF GEARING Friction and Knuckle Gearing. In machinery it is frequently necessary to transmit power from one shaft to another near to it. For this purpose gears are generally employed. Let a and 6, Fig. 156, be two such shafts. If now disks c and d are mounted upon these shafts, of such diameters as FIG. 156. -Friction Wheels. FIG. 157. -Knuckle Gears. to give the required speed ratio, we will have gearing in its simplest form. Such disks, having their edges covered with leather or other equiva- lent material, are called friction gears and are sometimes employed on light work. At best, how- ever, they will transmit but little power. If now we make semi -circular projections at equal distances apart upon the outside of the cir- cles c and d, and cut out corresponding depressions inside of the circles, as shown in Fig. 157, we will have a simple form of toothed gearing and the cir- 190 GENERAL PRINCIPLES OF GEARING 191 cles c and d will be the pitch circles. Such gears, called knuckle gears, are sometimes employed on slow-moving work where no special accuracy is required. They will not transmit speed uniformly. If the driver of such a pair of gears rotated at a uniform rate, the driven gear would have a more or less jerky movement as the successive teeth came into contact, and if run at high speed they would be noisy. Various curves may be employed to give to gear teeth such an outline that the driver of a pair of gears will impart a uniform speed to the driven one, but in common practice only two kinds are used, the cycloidal, or, as it is sometimes called, epicycloidal, and the involute. Epicycloidal Gearing. Let the circles a, b and c, Fig. 158, having their centers on the same straight line, be made to rotate so that their cir- cumferences roll upon each other without slipping. If the circle c has tracing points 1, 2, 3 upon its circumference, and when we start to rotate the circles point 1 is half way around from the posi- tion in which it is shown, then in rotating the cir- cles sufficiently to bring the tracing points to the position in which they are shown, point 1 will trace the line 1 ' inwardly from the circle a, and the line 1 " outwardly from the circle b. Point 2 will trace the two lines which are shown meeting at that point, one inwardly from the circle a, and one outwardly from the circle 6. Point 3 will similarly trace the two lines which met at that point. Inas- much as these lines were traced simultaneously by points at a fixed distance apart, it is evident that if the circle c were to be removed, and the circles 192 SELF-TAUGHT MECHANICAL DRAWING a and b were rolled back upon each other, these lines would work smoothly together, being in con- tact and tangent to each other at all times upon the line of the circle c. If the circle c is now placed beneath the circle b in the position shown, and the three circles are rolled together as before, the tra- cing points would trace lines inwardly from 6, and FIG. 158. Principle of Epi- cycloidal Gearing. FIG. 159. Principle of Invo- lute Gearing. outwardly from a, which would also work together smoothly if the circle c were removed and the cir- cles a and 6 were rolled back upon each other. It is evident that as the three circles are rolled together the lines formed by the tracing points are the same as though either a or 6 were taken by itself, and the circle c were rolled either within or upon it, hence the lines formed by the tracing points are either epicycloids or hypocycloids as the case may be, and so could be formed by the GENERAL PRINCIPLES OF GEARING 193 plotting method described in the geometrical problems. If these two sets of lines are now joined together so that the lines which extend inwardly from a or b form a continuation of those which extend out- wardly and reverse curves are made at a distance from the first set equal to the thickness of a gear tooth, and they are the-n cut off at such a distance both outside and inside of the circles a and 6 as to give to the teeth the proper length, it is evident that we will have a pair of per- fectly working gears. The circles a and b would roll upon each other without slipping and hence would FlG ^.-Definitions' of be true pitch circles. The Gear Tooth Terms. teeth would work smoothly together in constant contact, the point of contact being always on the line of the generating circle. The length of the point of the gear tooth, that is the portion lying outside of the pitch line, is usually made one-third of the circular pitch^the latter being the distance between the teeth meas- ured from center to center on the pitch line. The distance below the pitch line is made somewhat greater for the sake of clearance. For the names of the various parts of a gear tooth see Fig. 160. Cast gears have some backlash between the teeth to allow for the roughness of the castings, as shown in Figs. 161 and 163. It is evident that if another circle, either larger or smaller, were substituted for b in Fig. 158, the 194 SELF-TAUGHT MECHANICAL DRAWING lines formed by the generating circle c either within or upon the circle a would remain unchanged. Or if a different circle were substituted for a, the curves formed within or upon 6 would remain un- changed. Hence it follows that all gears in the epicycloidal system, having their teeth formed by the same generating circle and made of the same FIG. 162.-Rack with Epicycloidal Teeth. FIG. 161. Gears with Epicycloidal Teeth. size, will work together correctly, or % as it is com- monly expressed, are interchangeable. In standard interchangeable gears the generat- ing circle is made one-half the diameter of the smallest gear of the set, which has twelve teeth. This smallest gear will have radial flanks, as that part of the working surface lying within the pitch line is called, because the hypocycloid of a circle formed by a generating circle of half its size will be a straight line passing through its center. Fig. 161 shows a portion of a pair of such gears, Fig. 162 showing the rack. GENERAL PRINCIPLES OF GEARING 195 Gears with Strengthened Flanks. A further ex- amination of Fig. 158 will show that the curves formed by the generating circle when it is in the upper of the two positions in which it appears, work together by themselves, and those formed when it is in the lower position work similarly, so that it is not necessary that the same sized gener- r\ \J FIG. 164. Rack with Involute Teeth. FIG. 163. Gears with Involute Teeth. ating circle should be used in both positions, unless the gears are to be members of an interchangeable set of gears. Advantage may be taken of this fact to strengthen the roots of the teeth in a pinion. If, for instance, in Fig. 161, a smaller generating circle were used in the upper position, the effect would be to broaden out the roots of the teeth in the pinion, and to correspondingly round off the points of the teeth of the other gear. Gears with Radial Flanks. Another modification which may be made is to have the teeth of both gears with radial flanks. If, for instance, in Fig. 161 a generating circle were to be used in the 196 SELF-TAUGHT MECHANICAL DRAWING lower portion, of half the pitch diameter of the large gear, the effect would be to give to that gear radial flanks, and to make the points of the teeth of the small gear broader in order to work properly with them. Then both gears would have radial flanks. Such gears have been considerably used. They are not as strong as gears of the standard shape, and the only advantage is that it is easier to make the pattern, the teeth being all worked out with a flat-faced plane; but as the teeth of in- volute gears, described in the next section, can be worked out in the same way, and as such gears are interchangeable, the advantage is obviously in favor of the involute system for such work. Involute Gears. In involute gears the working surfaces of the teeth are involutes, formed not upon the pitch circles, but upon base circles lying within the pitch circles and tangent to a line, called the line of action, which passes obliquely through the point where the pitch circles cross the line connecting their centers. Let a and b, Fig.- 159, be pitch circles, and let the line cd be the line of action. Then e and /, being made tangent to the line cd, will be the base circles upon which the involutes are to be formed. If now this line of action be considered as part of a thread which unwinds from one base circle and winds up on the other, as the pitch circles are revolved back and forth upon each other, then if tracing points were attached to the thread at points 1, 2, 3, 4, 5 and 6, these points would describe involutes outwardly from the base circles, which, being formed simul- taneously in pairs and each pair being formed by GENERAL PRINCIPLES OF GEARING 197 a common point, would work together smoothly like those formed by the generating circles of the epicycloidal system. That the base circles are of such size as to just pass the thread as the pitch circles roll upon each other is proven by the fact that their radii, gd and gi, and he and hi, the radii gd and he being made at right angles to the line of action, are corresponding sides of similar triangles, the segments into which the line of action is di- vided by the line of centers being the other sides, and hence have the same ratio. It would only then FIG. 165. -Modified Form of Involute Rack Teeth. be necessary to reverse the direction of the thread to get curves for the other side of the teeth, and to give to the teeth their proper length inside and outside of the pitch line to obtain a pair of cor- rectly working involute gears. That part of the tooth of an involute gear which may lie within the base line is made radial. In the standard interchangeable involute gears the line of action is given an obliquity of 15 degrees (cut gears, 14J degrees) . This angle may be readily obtained by the combination of the triangles resting against the blade of the T-square shown in Fig. 166. The point of contact of the 198 SELF-TAUGHT MECHANICAL DRAWING teeth is always upon the line of action and the push of one tooth against another is in its direc- tion, hence its name. The teeth of the 15-degree involute rack have straight sides, inclined to the pitch line at an angle of 75 degrees as shown in Fig. 164. This shape, however, is subject to a slight modification to avoid interference of the points of the teeth with the radial flanks of small gears. Interference in Involute Gears. The points c and d, Fig. 159, where the line of action is tangent to the base circles, are called the limiting points. If the involutes which spring from either base cir- cle are so long as to reach beyond these limits on the other base circle, they will interfere with the radial flanks of the mating teeth. At A; is shown an elongated involute interfering with the radial flank of the mating tooth. This is, of course, a highly exagger- ated case. The interfer- FIG. 166. Obtaining a 15- or 75-degree Angle by 30- and 45-degree Tri- angles. ence will occur sooner as the line of action is made to cross the line of centers at a less oblique angle, as in standard gears, and still earlier as the pitch circle b is made larger. In gearing of standard pro- portions, a gear of 30 teeth is the smallest that will work correctly with a straight toothed rack. In the gears shown in Fig. 163, the teeth of the large gear pass beyond the limiting point of the small gear, and hence, if made of , true involute shape, GENERAL PRINCIPLES OF GEARING 199 their extremities will not work properly with the flanks of the small gear. There are three methods available to overcome this interference. First, to hollow out the flanks of the teeth of the small gear. Second, to round off the points of the teeth of the large gear. This is the method usually adopted, in interchangeable gears, the point being rounded off enough to clear the flanks of the smallest gear of the set. Fig. 165 shows the teeth of the rack so corrected in larger scale. Third, to cut off that part of the tooth in the large gear which extends beyond the limiting point of the small gear. This is done in special cases. The Two Systems Compared. The great point in favor of epicycloidal gearing would appear to be in its freedom from interference. It is necessary, however, in order to have epicycloidal gears run well, to have the pitch circles of the two gears of a pair just coincide, as shown in Fig. 161; but with involute gears the distance between centers may be varied somewhat without affecting their smoothness of operation, though where the points of the teeth are rounded off to avoid interference, as previously explained, the amount of variation which can be allowed is not great. As no value has been given to the angle at which the line of action crosses the line of centers in Fig. 159, it is evident that whether the base circles are brought nearer together or are carried further apart, circles which might then be drawn through the point where the line of action crosses the line of centers, would roll upon each other while the base circles 200 SELF-TAUGHT MECHANICAL DRAWING passed the thread as before, and hence would be true pitch circles for the time being. The amount of backlash, that is, the space between the faces of the teeth, would vary, but the smoothness of operation would not be affected. This property of involute gears is very valuable in cases where the distance between centers is variable, as in rolling mill gearing. In such cases, however, interfer- ence must be avoided by the first of the three methods explained, that of hollowing out the flanks of the teeth of the mating gear. The epicycloidal system is the older of the two, and cast gears are still quite largely made to this system, there being so many patterns of that sys- tem on hand. But though the epicycloidal system once had the field to itself, the fact that the invol- ute system has so largely replaced it, having al- most wholly superseded it for cut gearing, shows the trend of modern practice. It is sometimes urged against the involute system that the thrust on the shaft bearings is greater than with the epi- cycloidal system, on account of the obliquity of its line of action. But though the line of action is at an angle to the direction of the motion of the teeth when they are on the line connecting their centers, it is a constant angle; while it is never less, it is never more. With the epicycloidal system, on the other hand, though the teeth of the driver give a square push to the teeth of the driven gear when they are in contact on the line of centers, yet the direction of this pushing action being on the line of the generating circle, is variable, so that when the teeth are first coming into contact with GENERAL PRINCIPLES OF GEARING 201 one another they have an obliquity of action fully as great, if not greater, than standard involute gears. For this reason such authorities as the Brown & Sharp Co., Grant and Unwin, do not con- sider this objection as being of great weight. Twenty-Degree Involute Gears. It has been al- ready shown how the teeth of epicycloidal gears may be considerably strengthened where it is not necessary to have them interchangeable. In invol- ute gearing, when a stronger gear is desired than the standard 15-degree tooth provides for, recourse may be had to increasing the obliquity of the line of action. This makes the tooth considerably broader at the base, and correspondingly narrower at the point. The angle usually adopted in such cases is 20 degrees, and some makers report an increasing demand for such gears. Shrouded Gears. -When it is desired to strengthen the teeth of cast gears without increasing their size, or without using any other than a standard shape or tooth, the practice of shrouding them is sometimes resorted to. This consists in casting a flange on one or both sides of the gear. Full shrouding consists in having the flanges extend to the points of the teeth as shown in Fig. 167 ; half shrouding is where the flanges extend only to the pitch line as shown in Fig. 168. When the two gears of a pair are of nearly equal size so that their teeth would be of about the same strength it would be natural to use half shrouding on both gears as shown. When, however, there is much difference in the size of the gears, as shown in Fig. 167, it would be 202 SELF-TAUGHT MECHANICAL DRAWING natural to use full shrouding on the small gear, as otherwise its teeth would be weaker than those of the large gear. Shrouding is estimated to strengthen the teeth from 25 to 50 per cent. FIG. 167. FIGS. 167 and 168. -Shrouded Gears. Bevel Gears. In cylindrical or spur gears the pitch surfaces are cylinders of a diameter equal to the pitch circle; in bevel gears the pitch surfaces are cones, having their apices coinciding. In designing a pair of bevel gears as shown in Fig. 169, the center lines ab and cd are first drawn, and the pitch diameters then laid out from these GENERAL PRINCIPLES OF GEARING 203 lines as indicated. From the point where the lines of the pitch diameters meet at e, a line is drawn to the point where the center lines intersect at k. This gives one side of the pitch cone of each gear and from this the other sides of the cones are FIG. 169. Bevel Gears. readily drawn. All lines of the working surfaces of the gears meet at the point h. To lay out the teeth, the line/gr is first drawn through the point e and at right angles to eh. This gives the outside face of the teeth, and the points /and g become the apices of cones upon the devel- opment of which the teeth are laid out. With cen- ters at /and g the pitch line developments ei and ej are drawn, and upon these lines the teeth are laid out the same as for ordinary gears. When the two gears of a pair are of the same size they are called miter gears. 204 SELF-TAUGHT MECHANICAL DRAWING Worm Gearing. In worm gearing, as shown in Fig. 170, a screw having its threads shaped like the teeth of a rack engages with the teeth of a gear having a concave face and teeth of such shape as to fit the threads of the screw. If the screw is single threaded, one rotation of it will cause the gear to revolve the distance of one tooth ; if double threaded, the gear will turn two teeth, and so on. In worm gearing, the worm wears much faster than the gear; it is, therefore, frequently made of FIG. 170. Worm and Worm-Gear. steel while the worm-wheel is made of bronze, to give the combination increased durability. In involute worm gearing interference is com- monly avoided by the last of the three methods already mentioned. The points of the thread of the screw in Fig. 170 project but little beyond the pitch line, the root spaces of the gear being made correspondingly shallow. At the same time, the points of the teeth in the gear are made long enough to preserve their total length the same as usual, and the depth of the screw thread inside the pitch Iin2 is made sufficient for clearance. But un- GENERAL PRINCIPLES OF GEARING 205 less the worm-gear has less than 30 teeth, the standard shape of tooth will be satisfactory. Circular Pitch. In designing gearing, the old method (the one which is given in the older trea- tises on the subject) is to use the circular pitch; that is, the distance between the teeth, measured from center to center on the pitch circle. This method has many disadvantages. For instance, if it is required to make a pattern of a gear to mesh with one already on hand, the natural thing to do in measuring up the old gear is to first guess at where the pitch line is, and then measure straight across from one tooth to the next. This leads to two errors in the result; first, the probably incor- rect location of the pitch line, and, second, the dis- tance measured is the chordal pitch instead of the circular pitch. A noisy pair of gears would quite likely be the result. Again, as the ratio between the circumference and the diameter of a circle is not an even num- ber, but a troublesome fraction, the use of the cir- cular pitch method will give the pitch diameter of the gear in inconvenient fractions of an inch, un- less an equally inconvenient circular pitch is used. This method has so many disadvantages that it has been largely replaced by the more convenient "diametral pitch'* method. For cut gears the dia- metral pitch method is used almost exclusively; but for cast gears there are so many patterns on hand, made by the circular pitch method, that that method is still used considerably on such work, especially on the larger sizes of gears. Where one is designing new work, however, 206 SELF-TAUGHT MECHANICAL DRAWING where no old gear patterns made by the circular pitch method are used, the diametral pitch method will be by far the most convenient to use, which- ever style of tooth, whether involute or epicy- cloidal, may be adopted. PITCH DIAMETERS OF GEARS FROM 10 TO 100 TEETH, OF 1-INCH CIRCULAR PITCH. No. of Teeth Diam. in Inches No. of Teeth Diam. in Inches No. of Teeth Diam. in Inches No. of Teeth Diam. in Inches 10 3.183 33 10.504 56 17.825 79 25.146 11 3.501 34 10.823 57 18.144 80 25.465 12 3.820 35 11.141 58 18.462 81 25.783 13 4.138 36 11.459 59 18.781 82 26.101 14 4.456 37 11.777 60 19.099 83 26.419 15 4.775 38 12.096 61 19.417 84 26.738 16 5.093 39 12.414 62 19.735 85 27.056 17 5.411 40 12.732 63 20.054 86 27.375 18 5.730 41 13.051 64 20.372 87 27.693 19 6.048 42 13.369 65 20.690 88 28.011 20 6.366 43 13.687 66 21.008 89 28.329 21 6.685 44 14.006 67 21.327 90 28.648 22 7.003 45 14.324 68 21.645 91 28.966 23 7.321 46 14.642 69 21.963 92 29.285 24 7.639 47 14.961 70 22.282 93 29.603 25 7.958 48 15.279 71 22.600 94 29.921 26 8.276 49 15.597 . 72 22.918 ' 95 30.239 27 8.594 50 15.915 73 23.236 96 30.558 28 8.913 51 16.234 74 . 23.555 97 30.876 29 9.231 52 16.552 75 23.873 98 31.194 30 9.549 53 16.870 76 24.192 99 31.512 31 9.868 54 17.189 77 24.510 100 31.831 32 10.186 55 17.507 78 24.828 When the pitch of a gear is given in inches or fractions of an inch, the circular pitch is always meant; as, for instance, where a gear is said to be of 1-inch pitch, or IJ-inch pitch. To get the pitch diameter in such a case, it is necessary to multiply GENERAL PRINCIPLES OF GEARING 207 this pitch by the number of teeth in the gear, and then divide this product by 3. 1416, the ratio be- tween the circumference and the diameter. For ascertaining the pitch diameter of gears when using the circular pitch, the accompanying table will save much time. If the gear is of any other than 1-inch circular pitch, multiply the diameter here given for the required number of teeth, by the circular pitch to be used. Proportions of Teeth. The proportions of the teeth of gears where the circular pitch method is used, are given slightly different by various writ- ers. The length of the teeth is entirely arbitrary and therefore this discrepancy is quite natural. It is also unimportant, excepting as uniformity is desirable. The proportions as given by Grant are as follows : The addendum and dedendum are each made one-third of the circular pitch; the clearance, the distance of the root line below the dedendum line, is made one-eighth of the adden- dum; the backlash, the space which is allowed be- tween the sides of the teeth in cast gears, is made about the same as the clearance. This presents the proportions in fractions which are convenient to use, and at the same time makes the proportions practically the same as those of the diametral pitch method. Cut gears are made without backlash. Diametral Pitch. In the diametral pitch method the gear is considered as having a given number of teeth for each inch of pitch diameter. Gears having three, four, or five teeth to each inch of their pitch diameters are said to be of three, four, or five pitch. With this method the addendum 208 SELF-TAUGHT MECHANICAL DRAWING (the distance which the teeth project beyond the pitch line) is made equal to one divided by the pitch, so that the addendum on gears of three, four or five pitch would be, respectively, one-third, one- fourth or one-fifth of an inch. The advantages of this method are numerous. To get the diametral pitch of a gear it is only necessary to divide the number of teeth by the pitch diameter, or to divide the number of teeth plus two, by the outside diameter. A complete set of rules, as well as formulas and examples for calculating spur gear dimensions, will be given in the next chapter. It is quite a common practice in figuring gears made by diametral pitch to give only the pitch and the number of teeth, as 4 pitch, 18 teeth, or 4 D. P., 18 T. The letters D. P. stand for diametral pitch, the letters P. D. standing for pitch diameter. The pitch diameter is then found by dividing the number of teeth by the diametral pitch. When this method is used, the circular pitch becomes of secondary importance, but may be found by di- viding 3.1416 by the diametral pitch. When the circular pitch is given and the diametral pitch is desired, divide 3.1416 by the circular pitch. The diameter of a gear, unless otherwise specified, is always understood to be the pitch diameter. With the diametral pitch method, the pitch diameter, unless in even inches, will be in fractions of an inch corresponding to the pitch, so that the frac- tional parts of the diameter of gears of three, four or five pitch, for instance, would be thirds, fourths or fifths of an inch. GENERAL PRINCIPLES OF GEARING 209 The Hunting Tooth. It is a common practice in making gear patterns to have the teeth of the two gears of a pair of such numbers that they do not have a common divisor. For instance, instead of having 25 and 35 teeth in the gears of a pair, one may give to one of them one more or one less tooth, so as to insure all of the teeth of one gear coming into contact with all of the teeth of the other as they run together. This practice is condemned by some, however, on the ground that if any of the teeth are of bad shape it would be better to confine their injurious action within as narrow limits as possible, rather than to have them ruin all of the teeth of the other gear ; but the shape of badly formed teeth should be corrected as soon as the error is discovered. Approximate Shapes for Cycloidal Gear Teeth. That part of the~cycloidal curve which is used in the formation of gear tooth outlines is so short that it may be replaced with a circular arc which will very closely approximate it, and such arcs are generally used in the practical construction of gear patterns. In the following is given a table of such arcs with the location of the centers from which they are struck. The center from which that part of the tooth lying outside of the pitch line is drawn, the face of the tooth, will be inside of the pitch line, while the center from which that part of the tooth lying inside of the pitch line is drawn, the flank of the tooth, will be outside of the pitch line. These radii and center locations were ob- tained directly from a set of tooth outlines of 3-inch circular pitch, formed by rolling a genera- 210 SELF-TAUGHT MECHANICAL DRAWING ting circle, drawn upon tracing paper, upon a set of pitch circles, correct rotation being assured by the use of needle points pricked through the gen- erating circle into the pitch circle, the needle points serving as pivots upon which the genera- ting circle was swung through short successive stages, the forward movements of the tracing point in forming the cycloidal curves being also pricked through. Needle points were also used in the instruments which were used for tracing this curve when the radius and center location were determined. CYCLOIDAL TOOTH OUTLINES Radii and center locations for one-inch circular pitch. For any other pitch multiply the given figure by the required pitch. Number of Teeth. Face Radius. Inside of Pitch Line. Flank Radius. Outside of Pitch Line. 12 0.625 ins. 0.016 ins. Radial 14 0.666 0.021 4.00 ins. 2.35 ins. 16 0.697 0.026 2.80 1.33 18 0.724 0.031 2.37 _ 0.96 20 0.750 0.036 2.14 0.73 25 0.802 0.042 1.91 0.58 30 0.844 0.052 1.79 0.48 40 0.906 0.062 1.64 0.375 60 0.958 0.083 1.50 0.29 100 1.010 0.095 1.33 0.21 200 1.040 0.120 1.23 0.177 Rack 1.080 0.127 1.08 0.127 If the diametral pitch method is being used, the corresponding circular pitch may be found by dividing 3.1416 by the diametral pitch, as already mentioned. Involute Teeth. The construction of a correct involute tooth outline is so simple a matter as to make the use of tables of approximate circular GENERAL PRINCIPLES OF GEARING 211 arcs .unnecessary. An involute may be formed by the plotting method given in the geometrical prob- lems, but in most cases it may be more readily formed by the use of a sharply pointed pencil guided by a strong thread as shown in Fig. 171, where ab represents the pitch line of a gear, and cd represents the base circle, having a number of pins stuck into it at short distances apart. The thread being doubled, forms a loop to hold the pen- cil point. The thread being drawn tightly around the pins, the pencil is swung outward from the FIG. 171. Laying out an Involute Gear Tooth. base circle, forming the required involute. When gears of over thirty teeth are to mesh into others of less than that number, it will be necessary to slightly round over the points of the teeth to avoid interference with the radial flanks of the mating gear. For this purpose use a radius of 2.10 inches divided by the diametral pitch, with a center on the pitch line as shown in Fig. 172. This radius, 2.10 inches divided by the diametral pitch, is the same as that given by Grant for rounding off the points of the teeth of racks ; but actual trial on teeth of large size shows it to be correct for gear wheels also, giving a curve which coincides very closely with the epicycloidal shape which the point 212 SELF-TAUGHT MECHANICAL DRAWING should have to work correctly with the radial flank of the mating gear. That part of an involute tooth lying within the base circle is made radial, as previously stated, and a good fillet should be drawn in at the root. For this purpose use a radius of one-twelfth of the circular pitch. A templet which is fitted to this FlG. 172. Modified Tooth Form to Avoid Interference. outline is used to finish the drawing, and to mark out the teeth on the pattern. On large work the size of the base circle may be obtained by calculation more readily than by the use of the triangle, as shown in Fig. 166. When the line of action has an obliquity of 15 degrees, the diameter of the base circle will be equal to 0.966 of the pitch diameter. For 20-degree invo- lute gears the diameter of the base circle will be 0.94 of the pitch diameter. With the 20-degree involute system the teeth of the rack have an inclination of 70 degrees to the pitch line. With this system there will be no ne- cessity for rounding off the points of the teeth of the rack or of a large gear unless it meshes with GENERAL PRINCIPLES OF GEARING 213 a gear of less than 18 teeth. When, to avoid inter- ference, it does become necessary to round off the points of the teeth of the rack or of large gears, the same radius, 2.10 inches divided by the diam- etral pitch, is to be used, as in the 15-degree system, the center being on the pitch line as before. Proportions of Gears. A somewhat common rule is to make the rim and the arms of about the same thickness as the teeth at the root, though some make the thickness of the rim equal to the height of the tooth ; and to make the diameter and length of the hub about equal to about twice the diameter of the shaft. On spoked gears, the rim is also stiff- ened by ribbing it between the arms. On a light gear mounted on a relatively large shaft it would be natural to lighten the hub somewhat. The width of t the face of cast gears is usually made from two to three times the circular pitch. The face of bevel gears should not exceed one-fifth of the diameter of the large gear, and the face of worm gears should not exceed one-half of the diameter of the worm. Strength of Gear Teeth. When a gear is to be designed for a given work, the first question is how large to make the teeth to give the required strength. On their size will also depend the gen- eral proportions of the gear. It is comparatively easy to determine the work which the teeth are doing, that is, the strain or load which they are bearing, when the power which the gear transmits is known. A horse-power being the power required to lift 33,000 pounds one 214 SELF-TAUGHT MECHANICAL DRAWING foot in one minute, the load on the teeth will be 33,000 multiplied by the horse-power which is being transmitted, and divided by the velocity of the pitch line of the gear in feet per minute ; or, what is the same thing, 126,050 multiplied by the horse-power, and divided by the product of the pitch diameter in inches multiplied by the number of revolutions per minute. This latter figure, 126,050, takes into account the fact that in the first case the velocity is expressed in feet, while in this case the diameter is in inches, and also the fact that the velocity is a factor of the circumference instead of the diameter. While the load on the teeth may be readily determined, the question of how large they should be made to bear it is one where authorities have differed very much on account of the number of factors involved. First of all is the question of the material, usually cast iron, which is a variable quantity, both on account of the nature of the material itself, different grades varying greatly as to strength, and the liability of defects in the cast- ing. Then there is the question of whether the load should be considered as divided between two or more teeth or carried by one tooth, or the cor- ner of a tooth. Then there is the nature of the work: whether, the load will be uniform or whether the teeth will be subject to severe strain or shock. There are questions of the shape of the tooth, and the velocity at which the gear is running, the teeth having greater strength at slow speeds than at high speeds due to the shocks accompanying high velocities. GENERAL PRINCIPLES OF GEARING 215 To show the different results given by different writers we may take the case of a gear 24 inches diameter, 2 inches circular pitch, 4 inches face, running at 100 revolutions per minute. A rule given by Box in his treatise on mill gearing, and quoted by Grant and Kent, would make the gear safe for 9.4 horse-power. The rule in Nystrom's Mechanics gives 12.2 horse-power. Rules by other writers, quoted by Kent, give results as follows: Halsey, 22.6; Jones & Laughlin, 35; Harkness, 38; Lewis, 65.2. The rule by Prof . Harkness is the result of investigations conducted by him in 1886. He examined a great many rules, largely, how- ever, for common cast gears. Mr. Lewis's method, the result of his investigations of modern machine molded and cut gears, though giving much higher results than the others, is said to have proved sat- isfactory in an extensive practice, and so may be considered reliable for gears which are so well made that the pressure bears along the face of the teeth instead of upon the corners. It is customary in calculating gears to proceed on the assumption that the load is borne by one tooth, and in ordinary work, the size of the tooth may be determined by the load it may safely bear per inch of face and per inch of circular pitch. In 1879, J. H. Cooper selected an old English rule giving the breaking load of the tooth as 2000 X pitch X face, which, allowing a factor of safety of 10, would give us a safe load of 200 X pitch X face. Kent says of this rule that for rough ordinary work it "is probably as good as any, except that the fig- ure 200 may be too high for weak forms of tooth, 216 SELF-TAUGHT MECHANICAL DRAWING and for high speeds. " Lewis also considers this rule as a passably correct expression of good gen- eral averages. The value given by Nystrom and those given by Box for teeth of small pitch, are so much smaller than those of other authorities that Kent says they may be rejected as giving unnecessary strength. Accepting the factor 200 as a good average would leave one room for the exercise of individual judg- ment for the particular case in hand. If the speed were slow and the teeth were of strong shape, as where both the gears of a pair, or all of the gears of a train, have a reasonably large number of teeth, a higher figure, perhaps 225 or more, might be taken; while if the speed were higher and one of the gears had but few teeth, giving them a weak form, or if they were to be subject to much vibration or shock, a lower figure, perhaps as low as 125, might be taken. To ascertain the horse-power safely transmitted by an existing gear, we would then multiply to- gether its diameter, pitch (circular) and face, taken in inches, and the number of revolutions per minute, and multiply their product by 200, or whatever figure is selected, and divide the total product by 126,050. This may, perhaps, be ex- pressed clearer, as follows: diam. X rev. X circ. pitch X face X 200 Horse-power= " The figure 200 would give to the 24- inch gear previously considered 30.5 horse-power. The fig- ure 125 would give 19.0 horse-power. GENERAL PRINCIPLES OF GEARING 217 To ascertain the size of the teeth to transmit a given horse-power we may transpose the above rule and say that the product of the pitch multiplied by the face would be equal to 126,050 multiplied by the horse-power, and divided by the product of the diameter in inches, the number of revolutions per minute, and 200, or the figure selected; that is: ~. , , , , 126^050 X horse-power Circ. pitch X face :=-- Assuming some pitch and dividing this result by it would give the breadth of face. A few trials will give the desired ratio between pitch and breadth of face. If one has a table of square roots at hand, the work may be simplified by assuming some desired ratio, when the pitch will be the square root of the quotient of this figure, pitch multiplied by the face, divided by the ratio. If, for instance, the pitch multiplied by the face were found to be 12, and we desired them to be in the ratio of 2J to 1, the pitch would be equal to the square root of the quotient of 12 divided by 2i, or 2. 191, which would be about the same as li diametral pitch. Example. Required the size of the teeth of a gear 18 inches in diameter, to run 120 revolutions per minute, which shall transmit five horse-power, allowing 200 pounds load per inch of face, and inch of pitch. Then : 126,050 X 5 630,250 Pitch X face =- = "432^000 = L46 nearly. A circular pitch of 0.785 inch, correspond- 218 SELF-TAUGHT MECHANICAL DRAWING ing to 4 diametral pitch, would give a breadth of face of about 15 inches. For bevel gears take the diameter and pitch at the middle of the face. Mr. Lewis's method differs from the preceding in that instead of using a single constant, as 200 pounds per inch of pitch and inch of face, two constants are used, one, Y, a factor of strength depending on the number of teeth in the gear, and another, S, a safe working stress for different speeds of the pitch line, in feet per minute. The values of these constants are given in the accom- panying tables. The rule to get the horse-power of a given gear is: TJ p = circ. pitch X face X velocity X S X Y 33,000 the velocity being that at the pitch line in feet per minute, and the values of S and Y being taken from the tables. The velocity is, of course, the diameter in feet X 3.1416 X number of revolu- tions. If the diameter were taken in inches then the total product would be divided 'by 12. The product of the pitch multiplied by the face, to determine the size of teeth to transmit a given power, would then be 33,000 X H. P. Circ. pitch X face = v .T ~ " ~- ^ velocity X S X Y. The calculation should be made for the gear of the pair or train having the fewest teeth, as it would be the weakest, unless it were made of some stronger material as steel, or unless it were GENERAL PRINCIPLES OF GEARING 219 WORKING STRESS, S, FOR DIFFERENT SPEEDS AT PITCH LINE IN FEET PER MINUTE, FOR CAST IRON. Speed. s. Speed. s. 100 or less 200 300 600 8000 6000 4800 4000 900 1200 1800 2400 3000 2400 2000 1700 shrouded. If made of steel S might be taken 2i times the tabulated values. As a gear with cut teeth has from two to three times the strength of one with cast teeth, because of the more perfect contact, Mr. Lewis's method might be adapted to common cast gears by taking the value of S at from one-half to one-third of the tabulated value. By so doing one could bring into the calculation the question of shape of teeth and FACTOR FOR STRENGTH, Y, TO BE USED IN LEWIS'S FORMULAS. 0.078 0.083 0.088 0.092 0.094 0.096 0.098 0.100 13 3 0.3 -S-i 0.067 0.070 0.072 0.075 0.077 0.080 0.083 0.087 0.102 0.104 0.106 0.108 0.111 0.114 0.118 0.122 0.090 0.092 0.094 0.097 0.100 0.102 0.104 0.107 43 50 60 75 100 150 300 Rack 0.126 0130 0.134 0.138 0.142 0.146 0.150 0.154 0.110 0.112 0.114 0.116 0.118 0.120 0.122 0.124 220 SELF-TAUGHT MECHANICAL DRAWING speed, which would be especially desirable if the speed were high or the teeth of weak form. Tak- ing S at one-half the tabulated value would give to the 24-inch gear previously considered about the same power as allowing 200 pounds per inch of pitch and face, which Mr. Lewis considers a fair value. With cast gears where interchange- ability is not a necessary feature, the teeth of a small gear could of course be considerably strength- ened in the manner previously indicated for epicy- cloidal gears; or the 20-degree system might be used if the teeth have the involute form. Thurston's Rule for Shafts. The size of shaft which the gear will require may be found by the rule given by Thurston. Multiply the horse-power to be transmitted by 125 for iron, or by 75 for cold rolled iron, and divide the product by the number of revolutions per minute. The cube root of the quotient will be the size of the shaft. . The size of gear to give a required speed may be readily determined from the fact that the prod- uct of the speed of the driving shaft multiplied by the size of the driving gear or gears, should be equal to the product of the speed of the driven shaft, multiplied by the size of the driven gear or gears. This, perhaps, may be made clearer by placing the driving members on one side of a line, and the driven members on the other side, as in the following example. A shaft making 75 turns per minute has on it a gear of 200 teeth. Required the size of gear to mesh with it which shall drive its shaft 120 GENERAL PRINCIPLES OF GEARING 221 revolutions per minute. Letting x represent the size of the required gear we have Rev. driving shaft = 75 Size driving gear = 200 x = size driven gear. 120 = rev. driven shaft. Then as the product of the numbers on one side of the line equals the product of those on the other side, 75 X 200 -5- 120 will give the value of x, the number of teeth in the driven gear. This method applies to a train of gears as well as a pair. CHAPTER XIII CALCULATING THE DIMENSIONS OF GEARS IN the previous chapter, the general principles of gearing have been explained. The three kinds of gearing most commonly in use, spur gearing, bevel gearing and worm gearing, have been touched upon, and the fundamental rules for the dimensions of gear teeth have been given. In this chapter it is proposed to give in detail the rules and formulas for these three classes of gears, so as to enable the student to calculate for himself any general problem in gearing with which he may meet. Spur Gearing. In the following, machine cut gearing is, in particular, referred to; but the gen- eral formulas are, of course, of equal value for use when calculating cast gears. The expressions pitch diameter, diametral pitch and circular pitch have already been explained, and rules have been given for transferring circular pitch into diametral pitch, and vice versa. These rules, expressed as formulas, would be: in which P = diametral pitch, and P'= circular pitch. Assume as an example that the diametral pitch 222 CALCULATING THE DIMENSIONS OF GEARS 223 of a gear is 4. What would be the circular pitch of this gear? Using the formula given, we have: pf = == 0.7854 inch. When the diametral pitch and the pitch diameter are known, the .number of teeth may be found by multiplying the pitch diameter by the diametral pitch, as already mentioned in the previous chap- ter. This rule, expressed as a formula, would be : N=PXD in which N = number of teeth, D = pitch diameter, and P = diametral pitch. Assume that the diametral pitch of a gear is 4 and the pitch diameter 6i inches. What would be the number of teeth in this gear? By inserting the given values in the formula above, we would have : N = 4 X 6i = 25 teeth. If the number of teeth and pitch diameter of the gear are known, and the diametral pitch is to be found, a rule and formula for this may be arrived at by merely transposing the rule and formula just given. The diametral pitch equals the number of teeth divided by the pitch diameter, or, expressed as a formula: in which P, N and D signify the same quantities as in the previous formula. 224 SELF-TAUGHT MECHANICAL DRAWING Assume, for an example, that the number of teeth in a gear equals 35 and that the pitch diam- eter is 3J inches. What is the diametral pitch? If we insert the known values in the given for- mula, we have : 35 P = ^f = 10 diametral pitch. 05 Finally, if the diametral pitch and the number of teeth are known, the pitch diameter is found by dividing the number of teeth by the diametral pitch, which rule expressed as a formula, would be: As an example, assume that the number of teeth in a gear is 58 and the diametral pitch 6. What is the pitch diameter of this gear? By inserting the known values in the formula, we find : D = 5|- =9.667 inches> If it now be required to find the outside diam- eter of the gear, that is, the diameter of the gear blank, we make use of the following rule : The outside diameter equals the number of teeth plus 2, divided by the diametral pitch. Expressed as a formula, this rule is: TV N+2 P in which D ' = outside diameter of gear, and N and P have the same significance as before. As an example, assume that the number of teeth CALCULATING THE DIMENSIONS OF GEARS 225 is 58 and the diametral pitch 6. By inserting these values in the formula, we find the outside diameter: rv 58+2 60 D' = - = -^-= 10 inches. b b When the pitch diameter and the diametral pitch are known, the outside diameter is found as follows: Add the quotient of 2 divided by the diametral pitch to the pitch diameter; the sum is the outside diameter. This rule, expressed as a formula, is: in which the letters have the same significance as before. Assume that the pitch diameter of a gear is 9.667 inches, and the diametral pitch 6. Find the out- side diameter. By inserting the given values in the formula, we have: D f = 9.667 + ~ = 9.667 + 0.333 = 10 inches. o By a transposition of the rule and formula just given, we find that the pitch diameter equals the outside diameter minus the quotient of 2 divided by the diametral pitch. This rule, written as a formula, is- D = u- Jr Assume that the diametral pitch of a gear is 8, and the outside diameter 12 inches. What is the pitch diameter? D = 12-~f-=12-i = 111 inches. o 226 SELF-TAUGHT MECHANICAL DRAWING When the number of teeth and outside diameter are known, the diametral pitch may be found by adding 2 to the number of teeth and dividing the sum by the outside diameter; or, expressed as a formula: N + 2 P = D'. If the number of teeth in a gear is 96 and the outside diameter is 14 inches, what is the diame- tral pitch? If the known values are inserted in the given formula, we have : -D 96 + 2 98 . , P ~TZ~ = IT = diametral pitch. When the outside diameter and the number of teeth are known, the pitch diameter may be found by multiplying the outside diameter by the number of teeth, and dividing the product by the sum of 2 added to the number of teeth; or, as a formula: ZXX N "WTz Find the pitch diameter for the gear having 96 teeth and an outside diameter of 14 inches. 14X96 1344 , ^ == 96T2~ ~98~ = 13.714 inches. When it is required to find the center distance C between two gears in mesh with each other, we must first know the pitch diameters of, or the number of teeth in, the two gears. The center CALCULATING THE DIMENSIONS OF GEARS 227 distance equals one-half of the sum of the pitch diameters of the two gears : D + d 2 in which D and d denote the pitch diameters in the large and small meshing gears, respectively. The pitch diameters of two gears equal 9. 5 and 7 inches, respectively. Find the center distance between them when in mesh. 9.5 + 7 16.5 . O = ~^ ~~^~ = o.Zo inches. The center distance is also equal to the sum of the numbers of teeth in the two gears divided by two times the diametral pitch; or, as a formula: 2P in which N and n denote the numbers of teeth in the meshing gears. As an example, assume that the number of teeth in each two gears equals 95 and 75. The diametral pitch is 10. What is the center distance? n 95 + 75 170 : 2~~X~10 = "20" We will now find the dimensions of the tooth parts. The addendum (see Fig. 160) equals 1 divided by the diametral pitch. Expressed as a formula: in which A = addendum. 228 SELF-TAUGHT MECHANICAL DRAWING What is the addendum or height above the pitch line of a 5 diametral pitch gear tooth? A =4-= 0.2 inch. o The dedendum (see Fig. 160) equals the ad- dendum. The clearance, c, equals 0.157 divided by the diametral pitch, or: c _0157 P. What is the clearance at the bottom of the gear tooth (see Fig. 160) of a 4 diametral pitch gear? c = ^p - 0.039 inch. The full depth of the tooth equals the sum of the addendum, dedendum, and clearance, or JL JL , 0.157 2J.57 P ' P P P in which d ' = full depth of gear tooth. What is the full depth of a 4 diametral pitch tooth? d , = = . a539inche The thickness of a cut gear tooth at the pitch line equals 1.5708 divided by the diametral pitch; or, as a formula : T 1.5708 ~1T in which T = thickness of tooth at pitch line. CALCULATING THE DIMENSIONS OF GEARS 229 What is the thickness at the pitch line of a 4 diametral pitch gear tooth? T == ~ == 0.3927 inch. As a general example, let it be required to de- termine the various dimensions for a pair of gears, the one having 36 and the other 27 teeth. The gears are of 8 diametral pitch. By using the formulas given, we have : For the larger gear : Pitch diameter = -5 = -5- = 4.5 inches. r o Outside diameter = ~~w^ - 86 J" 2 = 4.75 inches. Jr o For the smaller gear: n 27 Pitch diameter = = - = 3.375 inches. Jr o Outside diameter = 5~~ = o = 3.625 inches. Jr o For both gears : Addendum = ~ = - - = 0.125 inch. f O Dedendum = ^ = -5- = 0.125 inch. Jr o ~ 0.157 0.157 ftmofi - v. Clearance = =r~ = ^~~ = 0.0196 inch. Full depth of tooth = = -~ - 0. 2696 inch. X O 36 + 27 63 015 . Center distance = - - 2x g == 16 = 3|| inch. 230 SELF-TAUGHT MECHANICAL DRAWING This concludes the required calculations neces- sary for a pair of spur gears. Bevel Gears. Bevel gears are used for trans- mitting motion between shafts whose shafts are not parallel, but whose center lines form an angle with each other. In most cases this angle is a --o FIG. 173. Diagram for Calculation of Bevel Gearing. right, or 90-degree, angle. The formulas for the dimensions of bevel gears are not as simple as those for spur gears, and an understanding of the trigonometrical functions, explained in Chapter VII, is necessary, as well as the use of trigonomet- rical tables. As bevel gears with a 90-degree angle between their center lines are the most common, CALCULATING THE DIMENSIONS OF GEARS 231 formulas will be given for this case only, in the following. In Fig. 173 a pair of bevel gears are shown, the dimensions of which are to be determined. The letters in the formulas below denote the following quantities : P = diametral pitch, Di = pitch diameter of large gear, D 2 = pitch diameter of small gear, 01 = outside diameter of large gear, 2 = outside diameter of small gear, NI = number of teeth in large gear, AT 2 = number of teeth in small gear, NI = number of teeth for which to select cut- ter for large gear, N 2 ' = number of teeth for which to select cut- ter for small gear, flu &i, Ci, 02, &2, c 2 , d and e = angles as shown in Fig. 173. A = addendum, A + C = dedendum = addendum plus clearance. If the pitch diameter and diametral pitch are known, the number of teeth equals the pitch diameter multiplied by the diametral pitch, or: N, = D, X P N 2 = D 2 X P If the number of teeth and the diametral pitch are known, the pitch diameter equals the number of teeth divided by the diametral pitch, or: 232 SELF-TAUGHT MECHANICAL DRAWING Angles a t and a 2 can be determined if either the numbers of teeth or the pitch diameters of both gears are known. The tangent for these angles, the pitch cone angles, equals the number of teeth in one gear divided by the number of teeth in the other, or the pitch diameter in one gear divided by the pitch diameter in the other, according to the following formulas : 2 , tan a 2 = ^ = ^ Angle a 2 also equals 90 - a lt The outside diameter equals the pitch diameter plus the quotient of 2 times the cosine of a t or a 2 , respectively, divided by the diametral pitch, or: 0,= Angles d and e are determined by -the formulas: , 2 sin a,! 2 sin a 2 tancZ= ^~ -~ 2.314 sin a^ 2.314 sin a 2 tan e -^r- ^ Angles 6 t , Ci, 6 2 and c 2 are determined by the formulas : &! = O-! + d Ct = di e & 2 = a 2 + d c = a - e CALCULATING THE DIMENSIONS OF GEARS 233 The number of teeth for which the cutter for cutting the teeth should be selected is found as follows : COS cosa 2 Finally the addendum, dedendum and clearance are found as in spur gears. As a practical example, assume now that two bevel gears are required, 8 diametral pitch, with 24 and 36 teeth, respectively. Find the various dimensions. n Ni 36 D 1 = -p -- ~^ =4.5 inches. N 2 24 D 2 = -p- = -g- = 3 inches. tan ttl == ^ == H = 1.5; a, = 56 20'. tan a 2 = ^ = ^ = 0.667; a 2 = 33 40'. JMi ob Qi=Di+ 2^0, =45+ 2X|554 = 46W incheg _ 2 = Z) 2+ ^f^ = 3 + ^^2 _ 3 m incheg _ -L O tan d = 234 SELF-TAUGHT MECHANICAL DRAWING b, = a, + d = 56 20' 4- 2 40' = 59 0'. Cl = a,- e = 56 20'-30' = 5320'. 6 2 = a 2 + d = 33 40' + 2 40' = 36 20'. c 2 = a2 - e = 33 40 ' - 3 ' = 30 40 '. ZV'= N * = - 36 1 COS ttj AT 2 ' = 0.554 N 2 24 cos a 2 0.832 = 65 approximately. = 29 approximately. A - -4- - 4- - 0.125 inch. Jr o = 00196inch o Whole depth of tooth = y 4- y + 0.2696 inch. Worm Gearing. Worms and worm gears are used for transmitting power in cases where great FIG. 174. -Worm. reduction in velocity and smoothness of action are desired. They are also used when a self -locking CALCULATING THE DIMENSIONS OF GEARS 235 power transmission is desirable, that is, when it is required that the mechanism itself, due to the friction between the worm and worm-wheel, should support the load without slipping if the driving power be rendered inoperative. In Figs. 174 is shown a worm and in Fig. 175 a worm-wheel; the dimensions to be found are, in most cases, given in these illustrations. The fol- lowing notation has been used in the formulas given below for worm and worm-wheels : P = circular pitch of worm-wheel = pitch of the worm thread, N = number of teeth in worm-wheel, Z>! = pitch diameter of worm-wheel, DT = throat diameter of worm-wheel, 01 = outside diameter of worm-wheel (to sharp corners) , R = radius of worm-wheel throat, C = center distance between worm and worm- wheel axes, D 2 = pitch diameter of worm, 2 = outside diameter of worm, D R = root diameter of worm, A = addendum, or height of worm tooth above pitch line, d = depth of worm tooth, a = face angle of worm-wheel. If the pitch of the worm and the number of teeth in the worm-wheel are known, the pitch diameter of the worm-wheel may be found by multiplying the pitch of the worm by the number 236 SELF-TAUGHT MECHANICAL DRAWING of teeth, and dividing the result by 3.1416, or, as a formula : D - 3.1416 The outside diameter of the worm, 2 , is usually assumed. To find the pitch diameter of the worm, the addendum must first be found. The addendum equals the pitch of the worm thread multiplied by 0.3183, or: A = P X 0.3183. Now the pitch diameter of the worm equals the outside diameter minus 2 times the addendum, or: > 2 = 2 - 2A. The root diameter of the worm can be found first after the full depth of the worm-wheel thread has been found. The full depth of the worm-wheel thread equals the pitch multiplied by 0.6866, or: d = PX 0.6866. Now the root diameter of the worm thread equals the outside diameter of the worm minus 2 times the depth of the thread, or: D R = 2 - 2d. FIG. 175. Worm-wheel. CALCULATING THE DIMENSIONS OF GEARS 237 The throat diameter of the worm-wheel is found by adding 2 times the addendum of the worm thread to the pitch diameter of the worm-wheel, or: D T = A + 2A. The radius of the worm-wheel throat is found by subtracting 2 times the addendum from the outside diameter of the worm divided by 2, or: R = Y 2 - 2A. The outside diameter of the worm-wheel (to sharp corners) is found by the formula below : G! = D T + 2 (R - tfcos- The angle a is usually 75 degrees. Finally, the center distance between the center of the worm and the center of the worm-wheel equals the sum of the pitch diameter of the worm plus the pitch diameter of the worm gear, and this sum divided by 2, or: Find, for an example, the required dimensions for a worm and worm-wheel, in which the worm- wheel has 36 teeth, the pitch of the worm thread is J inch, and the outside diameter of the worm is 3 inches. We have given P = J; N = 36; 2 = 3. X 36 3.1416 3.1416 = A = P X 0.3183 = i X 0.3183 = 0.15915 inch. D 2 = 2 - 2A = 3 - 0.3183 = 2.6817 inches. 238 SELF-TAUGHT MECHANICAL DRAWING d = P X 0.6866 = i X 0.6866 = 0.3433 inch. D R = 2 ~ 2d = 3 - 0.6866 = 2.3134 inches. D T = D, + 2A = 5.730 + 0.3183 = 6.0483 inches. R=^-2A=~- 0.3183 = 1.1817 inch. 0,= D T + 2 (R - R cos y) == 6.0483 + 2 X (1.1817 - 1.1817 X cos 37 30 X ) = 6.5375 inches. r A+A 5.730 + 2.6817 C = ~ - = 4.2058 inches. CHAPTER XIV CONE PULLEYS WHEN it is desired to have a variable speed ratio between two shafts which are belted together, the method of having reversed conical cylinders or drums mounted on the shafts, as shown in Fig. 176 and 177, is sometimes used. These permit any FIG. 176. -Simplest Form of ' ' Cone- Pulley." FIG. 177. An Im- proved Form of "Cone-Pulley." FIG. 178. The Mod- ern Type of Stepped Cone Pulley. desired change of speed, but they have disadvan- tages which on most work offset this advantage. It would be necessary, in the first place, to use a narrow belt to avoid undue stretching at the edges. Then, as the tendency of a belt is to mount to the largest part of a pulley, this tendency, acting in 239 240 SELF-TAUGHT MECHANICAL DRAWING the same way on the cones, would produce undue tension on the belt. If a crossed belt is used on such cones their faces would be made straight, as the belt would be equally tight in any position. This may be seen by an inspection of Fig. 179, where circles A and B represent sections of such cones on one line, and circles C and D represent sections on another line. If the cones have the FIG. 179. Diagram Showing relative Influence of Open and Crossed Belt on Pulley Sizes. same taper it is evident that the circle D will be as much larger than B as C is smaller than A, the gain in one diameter being offset by the loss in the other. Then, as the circumferences of circles vary directly as their diameters (the circumference of a circle having twice the diameter of another, for instance, will be twice as long as the circumfer- ence of the other) , whatever is gained on one cir- cumference_will be lost on the other. For a crossed belt then, it is only necessary that the cones have the same taper. When, however, an open belt is used, it becomes necessary to have the cones slightly bulging in the CONE PULLEYS 241 middle as shown in Fig. 177. By again inspecting Fig. 179 it will be seen that it is only when the belt is crossed that one cone gains as fast in size as the other loses, because it is only when the belt is crossed that the arc of contact of the belt on the pulleys is the same on all steps of the cone. In practice these cones are usually replaced by stepped or cone pulleys as shown in Fig. 178, so as to avoid the troubles with the belt previously mentioned. Applying the principles mentioned to cone pul- leys, we see that when a crossed belt is used, all that is necessary is that the sum of the diameters of any pair of steps shall be equal to the sum of the diameters of any other pair of steps. For instance, the sum of the diameters of steps 1 and r must be equal to the sum of the diameters of steps 2 and 2'. When, however, an open belt is used, as is usually the case, the sum of the diam- eters of the steps at or near the middle of the cone will have to be somewhat greater than the sum of the diameters of those at or near the ends. What is generally considered to be the best method of determining the size of the various steps of cone pulleys is that given by Mr. C. A. Smith in the ' 'Transactions of the American Society of Mechanical Engineers, ' ' Vol. X, page 269. Make the distance C, Fig. 180, equal to the distance between the centers of the shafts, and draw the circles A and B equal to the diameters of a known pair of steps on the cones. At a point midway between the shaft centers erect the perpendicular ab. Then, with a center on ab at a distance from 242 SELF-TAUGHT MECHANICAL DRAWING a equal to the length of C multiplied by 0.314, draw the arc c tangent to the belt line of the given pair of steps. The belt line of any other pair of steps will then be tangent to this arc. If the angle which the belt makes with the line of centers, de, exceeds 18 degrees, however, a slight modification of the above is made as follows : Draw a line tangent to the arc at c at an angle of 18 degrees with de', and with a center on a&, at FIG. 180. Method of Laying out Cone Pulleys. a distance from a equal to the length C multiplied by 0.298 draw an arc tangent to this 18-degree line. All belt lines which make an angle with de greater than 18 degrees are made tangent to this new arc. The sizes of the steps so obtained maybe verified by measuring the belt lengths of each pair. For this purpose a fine wire may be used, the wire being held in place by pins placed at close intervals on the outer half circumference of each pulley of the pair. CHAPTER XV BOLTS, STUDS AND SCREWS SCREWS for clamping work together are of three classes: through bolts, Fig. 181; studs, Fig. 182; cap screws, Fig. 183. In Fig. 181 the bolt is put entirely through both of the two pieces to be FIG. 181. Through Bolt for Holding two Pieces to- gether. FIG. 182. Stud used for Clamping one Piece to another. clamped together, and a nut is put onto the threaded end. This is considered to be the best method on cast iron work, both as regards efficiency and cheapness, as there is no tapping of any holes 243 244 SELF-TAUGHT MECHANICAL DRAWING in the cast iron. A tapped hole in cast iron is to be avoided, if possible, as, on account of the brittle nature of the material, the threads are liable to crumble or wear away easily. In many cases, however, it is not practicable to avoid tapping holes in cast iron, or questions of appearance may make the broad flange which is necessary when through bolts are used, undesirable. In such cases studs should be used. A stud consists of a piece of round stock threaded on both ends, and having a plain portion in the middle. The studs are screwed firmly into the tapped holes, which should be deep enough to prevent the studs from bottoming in them, the studs instead binding or coming to a bearing at the end of the threaded portion. The loose piece is then put on over the studs, and is held in place by the nuts. By using studs, any further wear of the tapped hole is avoided, as, when removing the loose part, the nuts only are taken off, the studs being left in the body piece. When the material of the parts which are being clamped together is of such a nature that threads formed in it are not liable to crumble or to rapid wear, then cap screws, Fig. 183, may be used to advantage. They give a neat appearance to a piece of work, and the nut is entirely eliminated. FIG. 183. Cap Screw used for Clamping Purposes. BOLTS, STUDS AND SCREWS 245 United States Standard Screw Thread. The most commonly used of all screw threads is the United States standard thread. A section, indicat- ing the form of this thread, is shown in Fig. 184. The thread is not sharp neither at the top nor at the bottom, but is provided with a flat at both of these points, the width of the flat being one-eighth of the pitch of the thread. The sides of the thread x^ FIG. 184. Form of the United States Standard Thread. form an angle of 60 degrees with each other. The "pitch" and the "number of threads per inch" should not be confused. The pitch is the distance from the top of one thread to the top of the next. If the number of threads is 8 per inch, then the pitch would be 4 inch ; and the flat on the top of a United States standard thread, which, as men- tioned, is one-eighth of the pitch, would be 1-64 inch. If the number of threads per inch is known, the pitch may be found by dividing 1 by the num- ber of threads per inch, or No. of threads per inch. If, again, the pitch is known and the number of threads per inch required, then No. of threads per inch = p. , 246 SELF-TAUGHT MECHANICAL DRAWING U. S. STANDARD SCREW THREADS. BOLTS AND THREADS HEX. NUTS AND HEADS. SQUARK NUT AND HEAD. 8 - 1 ll *o . 'o i-s 2 M o 3 || 0) c 1- 8-s 1 3 11 J3 I '^ si P <3 &*l ^S T3 Across Coi Rougl [o J |l v j 20 0.185 0.0062 0.049 0.027 ft 1? J ft If JL 18 0.240 0.0074 0.377 0.045 3"f H 11 Tti T 5 <> i || | 16 0.294 0.0078 0.110 0.068 :.| f a fl 1 ft ft 14 0.344 0.0089 0.150 0.093 If If ft 1 l/ 13 0.400 0.0096 0.196 0.126 1 If i ^> Iff ft 12 0.454 0.0104 0.249 0.162 II H T^T f 11 0.507]0.0113 0.307 0.202 1ft I s2 1/2 | T 9 6 1^ 10 0.620 0.0125 0.442 0.302 lj 3 G | H HI 9 0.731 0.0138 0.601 0.420 1ft If 9 If 2 | 11 2A 1 8 0.837 0.0156 0.785 0.550 If li 1 ii 2|f H 7 0.940 0.0178 0.994 0.694 144 1| 2^ 3 7 H 1ft 2 T 9 * H 7 1.065 0.0178 1.227 0.893 2 IT 6 2ft H l T 3 g m 6 1.160 0.020S 1.485 1.057 2ft 2* if It 5 ? 3ft if 6 1. 28410.0208 1.767 1.295 2f 2| H 1ft 3f| if 5 1.3890.0227 2.074 1.515 2ft 2i 6 2H if i T 9 ; 3f 5" 1.4910.0250 2.405 1.746 2| 2fi 3ft if IB 3 |j 1J 5 1.6160.0250 2.761 2.051 2H 2| 3JH 11 IB 43- 5 2" 2 4* 1.7120.0277 3.142 2.302 3ft 3f 2 Ifo 4 1| 21 4* 1.9620.0277 3.976 3.023 3} 3ft 4ft 21 2r 3 o 4fi 2i 4 2.176|0.0312 4.909 3.719 3 f 3rt 4i 2ft 5fl 2! 3 4 3* 2.4260.0312 2.6290.0357 5.940 7.089 4.620 5.428 4f 4^ 4.11 5| 2| 2 i-i 6 6H 31 3* 2.879 0.0357 8.296 6.510 5 4rl 5H 31 3j 3 G 7ft 3| 31 3 3.100 3.317 0.0384 0.0413 9.621 11.045 7.548 8.641 5| 51i 6 A 6H 3| 3H P 4 3 3.567 0.041312.566 9.963 6* 6ft 7 4 3}| 8*1 41 2J 3.798 0.043514.18611.32) 6ft 41 4 T 3 g Q_3_ 1 6 4* 4| 21 2f 4.028 4.256 0.0454 0.0476 15.90412.753 17.72114.226 71 s}| 4V 4f til 9| 5 2* 4.480 0.0500 19.635,15.763 7f 7ft 8fl 5 4fti lot! 2-i 4.730 0.0500 21. 648' 17. 572 8 7yf 9s 9 , 51 5y 3 g nil 1 6 2f 2| 21 4.953 5.203 5.423 0.052623.758 0.052625.967 0.055528.274 19.267 21.262 23.098 Ico oooo ot*-iHcooo|w 41 9f! 10ft 5i 5f II n| BOLTS, STUDS AND SCREWS 247 For example, assume that the pitch is 0.0625 inch. Then No. of threads per inch = . = 16. The accompanying table of United States stand- ard screw threads gives the standard number of threads per inch, corresponding to given diame- ters, the diameter at the root of the thread, the width of the flat at the top and bottom of the thread, the area of the full bolt body, and the area at the bottom of the thread. These dimen- sions are, of course, always the same with all manufacturers. As regards the sizes for hexagon nuts and heads, and square nuts and heads also given in the table, it may be said that all makers do not conform strictly to the sizes as given. The catalog of one large bolt manufacturing concern, which is at hand, gives the width across flats of finished bolt heads and nuts the same as the rough sizes given in the table, which, it will be seen, are founded on the rule that the width across the flats of the heads and nuts should equal one and one- half times the diameter of the body of the bolt, plus one-eighth of an inch. It will also be noticed that the thickness of the head or nut is the same as the diameter of the body of the bolt. With cap screws, although the length of the head is made the same as for bolts, or equal to the di- ameter of the bolt body, the diameter of the head, and the distance across flats, is made different as shown in table on the following page : 248 SELF-TAUGHT MECHANICAL DRAWING CAP SCREW SIZES. (From catalog of Boston Bolt Co. ) Size of Screw I T 7 * I T 5 6 i TV 1 A I A A & \- I f A it H f 1 t 1 1 i Width Across Flats Hex. Head i 1 i* i* it H Width Across Flats Square Head Check or Lock Nuts. When a bolt is subjected to constant vibrations there is a tendency for the nut to work loose. To overcome this tendency it is customary to employ a second nut, called a check or lock nut, which is screwed down upon the first one as shown in Fig. 185. When the first nut is screwed down to a bear- ing, the upper surfaces of its thread are in contact with the under surfaces of the bolt thread. When the check nut is screwed down, however, it forces the first nut -down so that the under surfaces of its thread come into contact with the upper thread sur- faces of the bolt. This means that the check nut has to bear the entire load. When, therefore, the two nuts are of unequal thickness, as is frequently the case, the thick nut should be on the outside. Bolts to Withstand Shock. When a bolt which is subjected to shocks fails, it breaks, of course, FIG. 185. Correct Arrange- ment when Using Check or Lock Nut. BOLTS, STUDS AND SCREWS 249 at the part having the least cross sectional area, that is, at the bottom of the thread. If now the body of the bolt be reduced so that its cross section is of the same area as the area at the bottom of the thread, a slight element of elasticity is intro- duced, and the bolt is likely to yield somewhat instead of breaking. This is considered very im- portant in some classes of work. The reduction of area may be accomplished by turning down the body of the bolt, or, according to some authorities, the same object is attained by removing stock from the inside by drilling into the bolt from the head end. Either method, it is stated, gives the same degree of elasticity to the bolt, but as the drilling method takes the stock from the center, the bolt is left stiffer to resist bending or twisting than when the stock is taken off the outside by turning. Wrench Action. When bolts or any form of screws are used to hold machine parts together, they must be strong enough not only to withstand the strain which is put upon them by the operation of the machine, but also to withstand the strain which is put upon them by the wrench in setting or screwing them up. In the case of a cylinder head, for instance, the strain upon the bolts due to the working of the engine will be the exposed area of the head, multiplied by the pressure per square inch. This divided by the number of bolts used will give the proportional part of this strain which each bolt must sustain. But in order to insure a tight joint, it is necessary that the bolts be not merely brought up to a bearing, but that they be 250 SELF-TAUGHT MECHANICAL DRAWING set up hard enough so as to press the cylinder and cylinder head surfaces firmly together. The force which the wrench exerts in doing this work will be equal to the circumference of the circle through which the hand moves in turning the wrench through one revolution, multiplied by the force in pounds exerted at the handle, and this product divided by the distance through which the nut advances in one revolution, that is, by the lead of the screw. This theoretical result is, of course, modified by the friction between the nut and the bolt, and between the nut and washer. In addition to this direct strain, there is also a twisting strain in the bolt, caused by the friction between the bolt and nut. To insure the bolts being sufficiently strong to resist these various forces, it is customary to make them somewhat more than double the strength that would be necessary to enable them to safely resist the pressure of the steam or other fluid in the cylinder; that is, they are made about double strength to enable them to resist the direct strain of the wrench action, and then this amount is in- creased about 15 or 20 per cent, to allow for the twisting action of the wrench. Allowing that a factor of safety of 4 would be sufficient to allow for the steam pressure only, a factor of safety of not less than about 9 or 10 would therefore be used to provide for the added strain on the bolt due to the wrench action. In the case of small bolts, where the workman might set them up much harder than is really necessary, a factor of safety of about 15 may be used. BOLTS, STUDS AND SCREWS 251 The distance apart which bolts can be spaced without danger of leakage is given by Prof. A. W. Smith as between 4 or 5 times the thickness of the cylinder flange for pressures between 100 and 150 pounds per square inch. In the case of bolts which are not under strain as a result of the wrench action, as in the case of FIG. 186. Example of Thread not under Stress due to Wrench Action. FIG. 187. Square Threaded Screw, such as is Generally used for Power Transmis- sion. the hook bolt shown in Fig. 186, a factor of safety as low as 4 might be properly used, if the load is steady. Assuming that the material of which the bolts are made has an ultimate strength of 40,000 to 60,000 pounds per square inch, the factors of safety previously indicated would give allowable working stresses of from 4000 to 15,000 pounds per square inch. 252 SELF-TAUGHT MECHANICAL DRAWING Screws for Power Transmission. In Fig. 187 is shown a square threaded screw such as is generally used for power transmission. In such a screw the depth of the thread is made one-half of the pitch. The size of the body of the screw, assuming that the work which the screw is doing brings a ten- sional stress on the screw, will be determined by the tensile strength of the material of which it is made and the factor of safety which is used. As a screw which is used for power transmission is subjected to constant wear when in use, the ques- tion of the proper amount of bearing surface in the threads of the nut is of first importance, in order that it may not wear out too rapidly. The area of the thread surface in the nut on which the pressure bears will be equal to the difference in area of a circle of a diameter equal to the outside diameter of the screw, and one of a diameter equal to the diameter at the root of the thread of the screw, multiplied by the number of threads ; or, letting D represent the outside diameter of the screw, and d represent the diameter of the body, the area will be: (D 2 - d 2 ) X 0.7854 X No. of threads in the nut. The allowable pressure per square inch of working surface will vary with the nature of the service required, whether fast or slow, and also with the lubrication, and with the material used. Where the speed is slow, say not over 50 feet per minute, and the service is infrequent, as in lifting screws, a pressure of 2500 pounds for iron or 3000 pounds for steel is allowable, while for more constant service some authorities limit the pressure to about 1000 pounds per square inch even when the BOLTS, STUDS AND SCREWS 253 lubrication is good. For high speeds a pressure of about 200 or 250 pounds is considered to be as much as should be allowed. For a screw which, fitting loosely in a well lubri- cated nut, is to sustain a load without danger of running down of itself, the pitch of the screw should not, according to Professor Smith, be greater than about one-tenth of its circumference. Efficiency of Screws. A square-threaded screw has a greater efficiency than a V- threaded one, as the sloping sides of the V-thread cause an increase of friction. Square threads are therefore preferable for power transmission. Experiments show that in the case of bolts used for fastenings, the friction of the nut on the bolt and washer may absorb 90 per cent, of the power applied to the wrench, leaving only 10 per cent, for producing direct com- pression. For square-threaded screws an efficiency of about 50 per cent, is considered fair, if the screws are well lubricated. Acme Standard Thread. While the square thread gives the greatest efficiency in a screw it is not as strong as one having sloping sides. Fig. 188 shows a section of a screw thread called the Acme or 29- degree thread, which is often used for replacing the square thread for many purposes, such as in screws for screw presses, valve stems, and the like. The use of such a screw permits the employ- ment of a split nut, when such construction is desirable, which would not be practicable with a perfectly square thread, and for this reason, as well as for the reason that it can be cut with greater ease than the square thread, it has of late 254 SELF-TAUGHT MECHANICAL DRAWING become widely used. In the Acme standard thread system the threads on the screw and in the nut are not exactly alike. A clearance of 0.010 inch is provided at the top and at the bottom of the thread, so that if the screw is 1 inch in diameter, for example, then the largest diameter of the thread in the nut would be 1.020 inch. If the root diam- eter of the same screw were 0.900 inch, then the smallest diameter of the thread in the nut would be 0.920 inch. The sides of the threads, however, fit perfectly. The depth of an Acme thread equals one-half the pitch of the thread plus 0.010 inch. The width FIG. 188. -Shape of Acme Screw Thread. of the flat at the top of the screw thread equals 0.3707 times the pitch; and the width of the flat at the bottom of the thread equals 0.3707 times the pitch minus 0.0052 inch. Miscellaneous Screw Thread Systems. Besides the screw thread systems already mentioned, a great many other systems are in more or less common use. Leading among these is the sharp V-thread, which, previous to the introduction of the United States standard thread, was the most com- monly used thread in this country. This thread is, theoretically at least, sharp at both the top and the bottom of the thread, the angle between the BOLTS, STUDS AND SCREWS 255 sides of the thread being the same as in the United States standard system, or 60 degrees. In ordinary practice, however, a small flat is provided on the top of the thread, because it would be almost impos- sible to commercially produce the thread otherwise ; and even if the thread could be produced, the sharp edge at the top would rapidly wear away. The sharp V-thread is being more and more forced out of use by the United States standard thread, although it must be admitted that it will probably long hold its own in steam fitting work, because of being especially adapted for making steam-tight joints. It answers this purpose probably better than any of the other common forms of threads. The Whitworth standard thread is not used to a very great extent in the United States, but it is the recognized standard thread in Great Britain. In this form of thread the sides of the thread form an angle of 55 degrees with each other, and the tops and bottoms of the threads are rounded to a radius equal to 0.137 times the pitch. This round- ing of the thread at the top provides for a thread which does not wear rapidly, and screws and nuts made according to this thread system will work well together in continuous heavy service for a longer period than would screws and nuts with any of the other standard thread forms. The fact that the threads are rounded in the bottom is advan- tageous on account of the elimination of sharp corners from which fractures may start. The main disadvantage of the thread, and the reason why the United States standard thread was adopted in this country in preference to the Whitworth stand- 256 SELF-TAUGHT MECHANICAL DRAWING ard, which is the older of the two, is to be found in the fact that it is more difficult to produce than a 60-degree thread with flat top and bottom. The Whitworth form of thread is used in this country mostly on special work and on stay-bolts for loco- motive boilers. A thread perhaps more commonly used than any of the others, with the exception of the United States standard thread, is the Briggs standard pipe thread, which is used, as the name indicates, for pipe fittings. This thread is similar to the sharp V-thread, having an angle of 60 degrees between the sides, and nearly sharp top and bottom ; instead of being exactly sharp at the top and bot- tom, however, it is slightly rounded off at these points. The difficulty of producing these slightly rounded surfaces has brought about a modification, at least in the United States, so that a small flat is made at the top, and the thread made to a sharp point at the bottom. It appears that a thread cut with these modifications serves its purpose equally as well as a thread cut according to the original thread form. Besides these systems, there are the metric screw thread systems. These use the same form of thread as the United States standard system, but the thread diameters and the corresponding pitches are, of course, made according to the metric system of measurement. Other Commercial Forms of Screws. Set-screws, shown in Fig. 189, are usually made with square heads, and have either round or cup-shaped points, and are generally case hardened. They are used BOLTS, STUDS AND SCREWS 257 for such work as fastening pulleys onto shafts, etc. Some set-screws are made headless, and are slotted for use with a screw-driver in places where it is undesirable that the _ screw projects beyond the work. The term machine screws covers a number of styles of small screws made for use with a screw-driver. Fig. 190 shows the principal styles. Machine screw sizes are usu- ally designated by numbers, the size and the number of threads per inch being usually given together, with a "dash" between ; thus a 10 24 screw would be a number 10 screw with 24 threads per inch. There are two standard systems for machine screw FlG. 189. Forms of Set-screws. FILLISTER HEAD FIG. 190. Forms of Machine Screws. threads, the old, which until recently was the only system, and the new, which was approved in 1908 by the American Society of Mechanical Engineers. The standard thread form of the old sy"stem was the sharp V-thread, with a liberal but arbitrarily 258 SELF-TAUGHT MECHANICAL DRAWING selected flat on the top. The basic thread form of the new system is that of the United States standard thread. The accompanying tables give the numbers and corresponding diameters and number of threads per inch of the old as well as the new system for machine screw threads. MACHINE SCREW THREADS, OLD SYSTEM. Number. Diameter. Threads per inch. Number. Diameter. Threads per inch. 1 0.071 64 12 0.221 24 1* 0.081 56 13 0.234 22 2 0.089 56 14 0.246 20 3 0.101 48 15 0.261 20 4 0.113 36 16 0.272 18 5 0.125 36 18 0.298 18 6 0.141 32 20 0.325 16 7 0.154 32 22 0.350 16 8 0.166 32 24 0.378 16 9 0.180 30 26 0.404 16 10 0.194 24 28 0.430 14 11 0.206 24 30 0.456 14 MACHINE SCREW THREADS, NEW SYSTEM. Number. Diameter. Threads per inch. I Number. Diameter. Threads per inch. 0.060 80 12 0.216 28 1 0.073 72 14 0.242 24 2 0.086 64 16 0.268 22 3 0.099 56 18 0.294 20 4 0.112 48 20 0.320 20 5 0.125 44 22 0.346 18 6 0.138 40 24 0.372 16 7 0.151 36 26 0.398 16 8 0.164 36 28 0.424 14 9 0.177 32 30 0.450 14 10 0.190 30 ! CHAPTER XVI COUPLINGS AND CLUTCHES A COUPLING is a device for connecting together the ends of two shafts or axles for the purpose of making a longer shaft, the term being usually limited to those devices which are intended for permanent fastening. The term clutch is used to designate a disengaging coupling. The simplest form of coupling consists simply of a sleeve or muff, made of a length about three times the diameter of the shaft, bored out to fit the shaft, and provided with a keyway its entire length, made to receive a tapering key. The ends of the shafting are, of course, also provided with keyways, and are inserted into the sleeve; then the key is driven in. In some couplings the sleeve is made tapering on the outside at both ends, and, being split, is clamped upon the shafts by means of rings or hollow conical sleeves which are driven onto the tapered ends, or drawn together by means of bolts. One of the most common forms of coupling is the flange coupling shown in Fig. 191. In this case a flanged hub is keyed to each of the shaft ends, and the flanges are then held together and prevented from turning relative to each other by bolts, as shown. In some cases the bolt heads and nuts are 259 260 SELF-TAUGHT MECHANICAL DRAWING provided with a guard by having the rim on the outer edge of the flange made deep as shown by the dotted lines on one side. This construction also allows the coupling to be used as a pulley, if necessary. In a coupling of this kind, the chief problem is to get the bolts of such size that their combined strength to resist the shearing action to which they are subjected equals the twisting FIG. 191. Flange Coupling. strength of the shaft. Letting d represent the diameter of the shaft in inches, its internal resist- ance to twisting is given by the formula ~ 5.1 in which T equals the internal resistance to twist- ing, or the twisting moment, and S the shearing strength per square inch of area in pounds. Regarding the shearing strength of materials Kent says : ' ' The ultimate torsional shearing re- sistance is about the same as the direct shearing resistance, and may be taken at 20,000 to 25,000 pounds per square inch for cast iron, 45,000 pounds COUPLINGS AND CLUTCHES 261 for wrought iron, and 50,000 to 150,000 pounds for steel according to its carbon and temper." The torsional and direct shearing resistance being the same, this quantity may be neglected if the shaft and coupling bolts are of the same material, and d s 5.1 the internal resistance factor or torsion modulus of the shaft, should be equal to the product of the radius of the bolt circle of the coupling, the number of bolts used, and the area of each bolt. Or, letting a represent the area of each bolt, R the radius of the bolt circle of the coupling, and n the number of bolts used, we would have : a = -~- + (R X ri). ). -L Example. Required the size of the bolts for a flange coupling for a 2-inch shaft. The radius of the bolt circle is 3 inches, four bolts being used. Using the notation in the formula given, our known values are: d = 2 inches, R = 3 inches, n = 4 bolts. If we insert these values in the formula we have: 03 o a = -g-y -^ (3X4)= g-^^ 12 =0.13 square inches. This area corresponds to a diameter of about A of an inch. To allow for the strain on the bolt caused by the action 01 the wrench, the next size 262 SELF-TAUGHT MECHANICAL DRAWING larger bolt, at least, or a i inch bolt, will be se- lected. The capacity of the bolt to resist shear- ing will be considerably increased by having the corners of the holes at those faces of the flanges which come together, somewhat rounded. If this is not done, the action of the flanges on the bolts will be like that of a pair of sharp shears. Experi- ments have shown that with the corners rounded, the capacity of the bolt to resist shearing may be increased 12 per cent. If the shaft and bolts are of different materials then the modulus ^ 5.1 should be multiplied by the shearing strength of the shaft in pounds per square inch and the product t= 3 d FIG. 192. Clamp Coupling. RXn should be multiplied by the shearing strength of the bolts per square inch, before dividing in the formula to get the bolt area. In Fig. 192 is shown another form of coupling much used. It consists of two parts bolted together over the joint in the shafting, a key and keyway being provided to prevent the slipping of the shafts. COUPLINGS AND CLUTCHES 263 By having a thickness of heavy paper interposed between the two parts of the coupling when it is bored out, it may be made to clamp very tightly onto the shafts. With either form of coupling, the length is made such that each shaft end is held by the coupling by a length of about one and one-half times its diameter, as indicated in the engravings. Oldham's Coupling. Fig. 193 shows a form of coupling which may be used for shafts which are FIG. 193. Oldham's Coupling. parallel, but slightly out of line. In this coupling each shaft end has a flanged hub attached to it. Across the face of each flange is planed a single groove passing through its center. Interposed between the two flanges is a disk, shown at the right, having tongues on both faces at right angles to each other, to engage in the grooves in the flanges. Hooke's Coupling or Universal Joint is used for connecting two shafts whose axes are not in line with each other, but merely intersect. The shafts A and B, and B and C, in Fig. 194, are thus con- nected by universal joints. If the shaft B is made telescoping, as is very often the case, a solid part 264 SELF-TAUGHT MECHANICAL DRAWING entering into and being keyed in a sleeve so as to prevent independent rotation, but yet permit a sliding action, then the two shafts A and C may move independently of each other within certain limits, the distance between their ends being capable of variation. The arrangement shown in Fig. 194 is used on various machine tools, notably on milling machines, flange drilling machines, etc. Many designs of flexible shafts are really only a combination of a great number of universal joints. FIG. 194. Application of Universal Joints and Telescoping When this coupling is employed for driving only one shaft at an angle with another, as if shaft A simply drove shaft B which, of course, is the fundamental type of universal coupling, then, if the driving shaft has a uniform motion, the driven shaft will have a variable motion, and so cannot be used in such cases where uniformity of motion of the driven shaft is necessary ; but where there are three shafts, as shown in the illustration, A will impart a uniform motion to C provided the axes of A and C are parallel with each other, as shown ; for if A, having a regular motion, imparts an irregular COUPLINGS AND CLUTCHES 265 motion to B, then if B, with its irregular motion, is made the driver, it will impart a regular motion to A, and as C is parallel with A it will also impart a regular motion to C. This form of coupling does not work very well if the angle a is more than 45 degrees. Clutches are of two general classes, toothed clutches and friction clutches. An example of a toothed clutch is shown in Fig. 195. In this clutch the part at the left is fastened to its shaft; the part at the right is free to slide back and forth upon FIG. 195. A simple Form of Toothed Clutch. its shaft, but is prevented from turning on the shaft by a key. The sliding motion for engaging or disengaging this part of the clutch is accom- plished by means of the forked lever and jointed ring, shown at the right, which latter engages in the groove A. Such a clutch, while giving a pos- itive drive, cannot, of course, be thrown in or out while the driving shaft is running at a high rate of speed. By having the back faces of the teeth beveled off as shown by the dotted lines, this diffi- culty is partly overcome, although the shock caused by the sudden engaging of the teeth still renders 266 SELF-TAUGHT MECHANICAL DRAWING the clutch unsuitable for operating at very high speed. To facilitate uncoupling, the driving faces may also be given an angle of about 10 or 12 degrees. Friction Clutches are generally made in one of the two styles shown in Figs. 196 and 197. The power which a clutch of the type shown in Fig. 196 will transmit, depends upon the power which is ap- plied to force the sliding part against the fixed part, FIG. 196. Friction -Disk Clutc.h. and the efficiency of the frictional force between the rubbing surfaces. As to the efficiency of the clutch, therefore, much depends upon the nature of the engaging surfaces, whether metal comes in contact with metal, or whether one of the surfaces has a facing of leather or wood. The efficiency is, of course, much increased by either a leather or wood facing. Professor Smith gives the efficiency of these different surfaces as follows: Cast iron on cast iron, 10 to 15 per cent. ; cast iron on leather, COUPLINGS AND CLUTCHES 267 20 to 30 per cent. ; cast iron on wood, 20 to 50 per cent. The horse-power which such a clutch will trans- mit will be found by multiplying the velocity of the parts in contact, in feet per minute, taken at their mean diameter as indicated at D, by the force which is being applied at this diameter in the direction of revolution, and dividing this product by 33, 000. The force which is acting at the diameter D to produce revolving motion is equal to the pres- sure which is being applied to force the two parts of the clutch together, multiplied by the coefficient of friction (as the frictional efficiency between the surfaces in contact, as given above, is called) of the materials which form the driving surfaces. Example. What power will a clutch of the type shown in Fig. 196 transmit if running at a speed of 250 revolutions per minute? The diameter D is 18 inches, and a pressure of 50 pounds is exerted to force the two clutch faces together. One of the clutch parts has a leather facing, and the coefficient of friction is 0.25. The general formula for finding the horse-power of a clutch of this type is: IT p >X3.1416XnXPX/ 33,000 in which H.P. = horse-power transmitted, D = mean diameter of friction sur- faces in feet, n = revolutions per minute, P = pressu/e between clutch surfaces in pounds, / = coefficient of friction. 268 SELF-TAUGHT MECHANICAL DRAWING The values to be inserted in the formula, which are given in this problem, are as follows : D = ]n = 1.5 foot, n = 250 revolutions, P = 50 pounds, / = 0.25. Inserting these values in the formula we have : 1.5 X 3.1416 X 250 X 50 X 0.25 n ,_ "33,000 The formula given may be transposed in various ways according to the requirements of the problem ; if, for instance, it is desired to know what pressure must be applied to transmit a given horse-power, then: H.P. X 33,000 D (in feet)" X 3.1416 X n X /. If the pressure is known, and it is required to find what diameter the clutch must be made to transmit a given power, then : D (in feet) - - ~ 3.1416 X n X P X /. If the pressure and diameter are both known, then the number of revolutions which the clutch must make per minute to transmit a given horse- power will be : H.P. X 33,000 n = D (in feet) X 3.1416 X P X /. It may be said that the capacity of the clutch to transmit power is independent of the area of the COUPLINGS AND CLUTCHES 269 friction surfaces; for, if the friction surface is increased the pressure which is applied to force the two parts of the clutch together is simply dis- tributed over a much greater area, giving a smaller pressure per square inch. The durability would be increased, but the horse-power capacity would re- main unchanged. The conical clutch shown in Fig. 197 may be made to run metal to metal, or the hollow part may FIG. 197. Friction Cone Clutch. be made larger to allow of the insertion of wooden blocks. This would increase the efficiency, but at the expense of the durability. The principle of this form of clutch may be explained by referring to the diagrammatical sketch at the right of Fig. 197, where the angle ACB represents the angle which the opposite sides of the clutch make with each other, the line DC representing the axis of the shaft. If now line bd of the small triangle abd be considered as representing the magnitude 270 SELF-TAUGHT MECHANICAL DRAWING of the force acting in the direction of the axis of the shaft to force the two parts of the clutch together, then if ab is at right angles to AC, ab will represent the resultant magnitude of the force acting on the face of the clutch at right angles to its surface, according to the principles explained in the chapter on the elements of mechanics. The efficiency of the clutch will therefore be as much greater than that of a flat-faced clutch as ab is greater than bd. The horse-power of such a clutch, using the same notation as before, would, therefore, be: _ D (in feet) X 3.1416 X n X P Xf v ab^ 33,000 K bd. But from the chapter on the solution of triangles we know that T = sine of angle bad. Hence ab 1 bd sin bad. But angle bad equals angle x, the angle which the conical surface of the clutch makes with the axis of the shaft. Therefore ab = 1 bd sin x and our original formula takes the form: TJ P D On feet) X 3.1416 X n X P Xf ' 33,000 X sin x. COUPLINGS AND CLUTCHES 271 Transposing this formula as before for the flat- faced clutch, gives us: H.P. X 33,000 X sin a? ~ D (in feet) X 3.1416 XnXf. nr - ,, H.P. X 33,000 X sin x D (in feet) - ~ " n = - = ,____^ srn^ D (in feet) X 3.1416 X Pxf. The sine of x may be taken from the tables of trigonometric functions previously given in the chapter on the solution of triangles, or it may be found by dividing the length bd (Fig. 197) by the length ab. The power necessary to force the two parts of the clutch together may be neglected, as the slip- ping which occurs as they are engaging allows them to come together with but little pressure beyond what is required for power transmission purposes. The angle which the face of the clutch makes with the shaft (the angle x in the diagram at the right in Fig. 197) should be such that the clutch does not grip too quickly when thrown into gear, nor require too much pull to release. Making this angle between 7 and 12 degrees conforms to the average given by different authorities. CHAPTER XVII SHAFTS, BELTS AND PULLEYS Shafts. The twisting strength of a shaft, as stated in the preceding chapter, is given by the formula T = 5.1 in which T = twisting moment, or force which acting at a distance of one inch from the center of the shaft would produce in it a torsional shearing stress of S pounds per square inch, d = diameter of shaft in inches, S = torsional shearing stress in pounds per square inch. Expressing this formula in words we may say that the cube of the diameter in inches multiplied by the torsional shearing stress, and this product divided by 5.1, gives the force which acting at a distance of one inch from the center of the shaft would produce in it the given torsional shearing stress. The twisting moment T equals, therefore, the force F l9 acting at a distance of one inch from the center of the shaft, times 1 ; it also equals any other force F exerting a twisting action on the 272 SHAFTS, BELTS AND PULLEYS 273 shaft multiplied by its distance from the center of the shaft. The formula given can hence be written 5.1 vn which F = any force acting at a distance r from the center of the shaft. Transposing this formula to obtain the distance from the center (r) at which a given force would have to act to set up a torsional shearing stress S in the shaft, we would have: r " = KlXF. The force which would be necessary to set up a stress S in the shaft when acting at a given distance would be : = 5.1 Xr. The diameter of shaft to resist a given force acting at a given distance would be: d -- ^F r X5.1 The torsional shearing strength of ordinary shafting is about 45,000 pounds to the square inch, and of steel shafting from about 50,000 to 150,000 pounds, according to its quality; these figures should be divided by five or six to give a safe working stress. The above formulas, however, are based on the assumption that the force acting is of a purely 274 SELF-TAUGHT MECHANICAL DRAWING twisting nature, as if a hand-wheel were put onto the end of the shaft, and the tendency to bend the shaft, caused by the pull of one hand, were counter- acted by the push of the other hand. In the case of a shaft actuated by a rocker arm, as sometimes occurs in machines, the tendency to bend the shaft caused by the push on the arm could be provided for by using a somewhat higher factor of safety. If the arm were placed at some distance from the bearing, however, the tendency to bend the shaft might be greater than the twisting effect. The methods of calculating the size of shafts for transmitting a given power, so as to take into account both the twisting and bending effects pro- duced by the pull of the belt are quite complicated, and the beginner will ordinarily find it best to use some of the empirical formulas for that purpose which are intended to take into account both of these effects. The following rules by Thurston are considered to afford ample margin for strength for shafts which are well supported against springing: To find the diameter of a cold rolled iron shaft to transmit a given horse-power, multiply the horse- power to be transmitted by 75, and divide the product by the number of revolutions per minute that the shaft is to make. The cube root of this quotient will be the diameter of the shaft. If the shaft is to be of turned iron, proceed as above, except that the horse-power to be trans- mitted is to be multiplied by 125 instead of 75. This rule is "for head shafts, supported by bear- SHAFTS, BELTS AND PULLEYS 275 ings close to each side of the main pulley or gear, so as to wholly guard against transverse strain. " If the main pulley is at a distance from the bearing, the size of the shaft will need to be increased, while for ordinary line shafting, with hangers 8 feet apart, the size may be reduced, figures of 90 for turned iron, and 55 for cold rolled iron shafting being substituted for those given in the rule ; or, in the case of shafting for transmission only, without pulleys, figures of 62.5 for turned iron, and 35 for cold rolled iron are substituted. To find the horse-power which a given shaft will transmit, multiply the cube of its diameter by the number of revolutions per minute, and divide the product by 125 for turned iron, or by 75 for cold rolled iron. For line shafting substitute the figures given by 90 and 55, respectively. The horse-power which is being transmitted is determined by multiplying the pull in pounds which the belt exerts (or the push which the teeth of the driving gear exert, if gears are used) by the diameter of the pulley in inches (or the pitch diameter of the gear in inches) and multiplying this product, again, by the number of revolutions per minute of the shaft; then divide this product by 126,050, and the quotient gives the horse-power transmitted. Expressed as a formula this rule would be: n P X D X N n '^' 126,050 276 SELF-TAUGHT MECHANICAL DRAWING in which P = pull on belt or push on gear teeth in pounds, D = diameter of pulley or pitch diameter of gear in inches, N = number of revolutions per minute of pulley or gear. Belts. The theoretical horse-power which a belt will transmit is equal to the pull which the belt exerts in pounds, multiplied by its velocity in feet per minute, and this product divided by 33,000. The question then arises as to what is the allowable stress to be put upon a belt. A common rule of practice for ordinary belting is that for single thickness belts the horse-power transmitted equals the breadth of the belt in inches, multiplied by its velocity in feet per minute, this product being divided by 1,000. This rule assumes a belt pull of 33 pounds per inch of width. Many authorities, however, would allow a much higher tension. The higher the tension, however, the narrower the belt for a given horse-power, and the greater the stretch, the more frequent the necessity for relacing, and the shorter the life of the belt. Allowing 33 pounds tension per inch in width for the thinnest commercial single belt, and allowing the tensions for increased thicknesses given by a large belt manufacturing concern, would give the following formulas for the transmission capacities of given belts: SHAFTS, BELTS AND PULLEYS 277 Single belt, A inch thick, H.P. = Breadth^velocity. Single belt, J inch thick, H.P. = Breadth X velocity. 800 Light double, # inch thick, H.P. = Breadth X velocity, 733 Heavy double, & inch thick, #. P. = Breadth X velocity. 687 Heavy double, A inch thick, H.P. = Breadth X velocity. 660 Heavy double, | inch thick, H.P. = Breadth X velocity. 550 Heavy double, if inch thick, H.P. = Breadth X velocity. 500 In these formulas the breadth of the belt is understood to be in inches, and its velocity in feet per minute, the letters H.P. meaning horse-power. Transposing the above formulas to ascertain the breadth of belt required to transmit a given power, we would have : Single belt, T \ inch thick, Breadth = H ' R x 100 Velocity Single belt, i inch thick, Breadth =. H - P - x 80 Velocity Light double, U inch thick, Breadth = H " P ' x 73B Velocity Heavy double, & inch thick, Breadth - H ' P ' x 687 Velocity Heavy double, T \ inch thick, Breadth = H ' P ' x 66 Velocity Heavy double, | inch thick, Breadth =- H ' P - x 55 Velocity Heavy double, ifinch thick, Breadth ^ H - P ' x 50Q Velocity 278 SELF-TAUGHT MECHANICAL DRAWING These formulas are all for laced belts. A belt made endless by being lapped and cemented or riveted is considered to be nearly 50 per cent, stronger than a laced belt, and is thus capable of transmitting nearly 50 per cent, more power; or the breadth of an endless belt to transmit a given power would not need to be more than between two- thirds to three-quarters of the breadth of a laced belt. Metal fastenings are not considered to make as strong a belt as lacings. If the foregoing formulas had been made on the basis of an allowable stress of 45 pounds for each inch in width of a single belt, a figure which many consider perfectly safe for a belt in good condition, they would have shown the belts as being capable of transmitting one-third more power than at 33 pounds stress per inch; to transmit a given power a belt would then need to be not more than three- quarters of the width. It will be seen from these formulas that the power transmitting capacity of a belt depends upon its breadth (a wide belt allowing an increased tension) or on its velocity. Increasing the width of the belt without increasing the tension to corre- spond would not give any increase of power trans- mitting capacity, as the given tension would simply be distributed over so much more pulley surface ; but a tight belt means more side strain on shaft and journal. Therefore, according to Griffin, from the standpoint of efficiency, use a narrow belt under low tension at as high a speed as possible. The de- sired high speed is, of course, secured by simply putting on large pulleys. SHAFTS, BELTS AND PULLEYS 279 Speed of Belting. The most economical speed is somewhere between 4000 and 5000 feet per minute. Above these values the life of the belt is shortened ; ' ' flapping, " ' ' chasing, ' ' and centrifugal force also cause considerable loss of power at higher speeds. The limit of speed with cast iron pulleys is fixed at the safe limit for the bursting of the rim, which may be taken at one mile surface speed per minute. The formulas given for the horse-power trans- mitted assume that the belt is in contact with just one-half of the pulley ; or, in other words, that the arc of contact is 180 degrees. If the arc of contact is increased, as it might be in the case of a crossed belt, until it becomes 240 degrees, or two- thirds of the circumference of the pulley, it is stated that the adhesion of the belt to the pulley, and conse- quently the efficiency of the belt, will be increased 50 per cent. If, on the other hand, the arc of con- tact should be reduced to 120 degrees, or one- third of the circumference of the pulley, as might be the case with open belts where the shafts were near together, and the pulleys were very unequal in size, the efficiency is stated to be only 60 per cent, of what the formulas would show ; if the arc of con- tact should be reduced to 90 degrees, the efficiency is stated to be only 30 per cent. From these percentages one can form a fairly good idea of what percentage to allow for varying arcs of contact. In most cases, however, it will probably be correct enough to assume the arc of contact to be 180 degrees. In all cases of open horizontal belting the lower 280 SELF-TAUGHT MECHANICAL DRAWING run of the belt should be made the working part, so that the sag of the upper run will increase the arc of contact. In the location of shafts that are to be connected with each other by belts, care should be taken to secure them at a proper distance from one another. It is not easy to give a definite rule what this dis- tance should be. Some authorities give this rule: Let the distance between the shafts be ten times the diameter of the smaller pulley ; but while this is correct for some cases, there are many other cases in which it is not correct. Circumstances generally have much to do with the arrangement; and the engineer or machinist must use his judgment, mak- ing all things conform, as far as may be, to general principles. The distance should be such as to allow a gentle sag to the belt when in motion. The Page Belting Co. states that if too great a distance is attempted, the weight of the belt will produce a very heavy sag, drawing so hard upon the shafts as to produce considerable friction in the bearings, while at the same time the belt will have an un- steady, flapping motion, which will destroy both the belt and the machinery. As belts increase in width they should be made thicker. It is advisable to use double belts on pulleys 12 inches in diameter and larger. If thin belts are used at very high speed, or if wide belts are thin, they almost invariably run in waves on the slack side, or "travel" from side to side of the pulleys, especially if the load changes suddenly. This waving and snapping that occurs as the belts straighten out, wears the belts very fast, and SHAFTS, BELTS AND PULLEYS 281 frequently causes the splices to part in a very short time, all of which is avoided by the employment of suitable thickness in the belts. The Page Belting Co. states that driving pulleys on which are to be run shifting belts should have a perfectly flat sur- face. All other pulleys should have a convexity in the proportion of about -fj of an inch to one foot in width. The pulleys should be a little wider than the belt required for the work. Pulley Sizes. The sizes of pulleys to give a re- quired speed, or the speed which will be obtained with given pulleys may be readily found from the fact that the product of the speed of the driving shaft, in revolutions per minute, and the diameters of all driving pulleys, on the main and on counter- shafts, multiplied together, will be equal to the product of the diameters of all driven pulleys and the speed of the last driven shaft, in revolutions per minute, multiplied together; so that if the size of one driven pulley, for instance, is required, its size may be found by dividing the product of the speed of the driving shaft and all driving pulleys multiplied together, by the product of speed of the final driven shaft and the diameters of such driven pulleys as are given, multiplied together. The re- sult will be the required pulley size. Example. A shaft making 200 revolutions per minute has mounted on it a pulley 18 inches in diameter which belts onto a 6-inch pulley on a countershaft. The countershaft has mounted on it a 20-inch pulley which belts to a pulley on the spindle of a machine which is to make 3000 revolu- tions per minute. What size pulley will be required on the spindle. CAH? rrv 282. SELF-TAUGHT MECHANICAL DRAWING Placing the speed of the driving shaft, and the sizes of all driving pulleys on one side of a vertical line, for convenience sake, and the sizes of all driven pulleys and the speed of the last driven shaft (or spindle) on the other side, and letting x represent the required size we would have: Speed of shaft = 200 Pulley on shaft = 18 Driving pulley on countershaft = 20 6 = Driven pulley on coun- tershaft. x = Required size of pulley on spindle. 3000 = Speed of spindle. i Then 200 X 18 X 20 = 6 X x X 3000 = 200J<_18_X_20 _ _72 1 000 6 X~3000 18,000 The diameter of the pulley on the spindle would therefore have to be 4 inches. If this size had been given, and the speed of the spindle had been required, x might have been taken to represent the required speed, when the same process would have given the desired information. Twisted and Unusual Cases of Belting. It frequently happens that, in transmitting power, conditions present themselves in which ordinary straight belting, either open or crossed, will not serve the purpose, and recourse must be had to some form of twisted belting, either quarter turn belting or belting guided by idler pulleys. In the following are given some of the principal con- ditions. Fig. 198 shows a quarter turn belt, by which power can be transmitted from one shaft to another at right angles to it. The condition necessary for SHAFTS, BELTS AND PULLEYS 283 the successful working of this arrangement is that the middle of the face of the pulley toward which FIG. 198. Arrangement of Pulleys for Quarter-Turn Belt. FIG. 199. Another Arrange- ment for Transmitting Power between Shafts at Right Angles. the belt is advancing shall be in line with the edge of the pulley that the belt is leaving. An exami- 284 SELF-TAUGHT MECHANICAL DRAWING nation of both the plan and elevation views will make this clear. While this is the simplest arrangement for this purpose, it has several drawbacks. The edgewise stress on the belt as it is leaving either pulley is very severe on the belt. It also causes a consider- able loss of contact with the pulley face, with corresponding loss of power transmission capacity. The edgewise stress also makes it necessary, if durability is to be considered, to have the belt relatively narrow. Incidentally, also, any reversal of the motion will cause the belt to immediately run off the pulleys. Fig. 199 shows another arrangement for trans- mitting power from one shaft to another at right angles to it, which overcomes all of the objections mentioned to the arrangement shown in Fig. 198, but at the expense of a double length belt and an extra pair of pulleys. As shown in the illustration, A and B are tight pulleys, and C and D are loose pulleys. The belt, as it leaves the tight pulley A, passes down under the loose pulley D, up over the loose pulley C, down under the tight pulley B, and then up over the tight pulley A, making a complete circuit. The loose pulleys, it will be seen, revolve in an opposite direction to the shafts on which they are mounted. Fig. 200 shows an arrangement by which, by employing loose guide pulleys, power may be trans- mitted from one shaft to another so close to it as to prohibit direct belting. If the main pulleys are of the same size, and their shafts are in the same plane, the guide pulleys may be mounted on a SHAFTS, BELTS AND PULLEYS 285 single straight shaft at right angles to a plane passing through the axes of the shafts on which the main pulleys are mounted. If, however, the main pulleys are of unequal size, as shown in the illustration, the guide pulleys will have to be in- clined to such an angle that the center of the face FIG. 200. Arrangement of Belt Transmission Using Loose Guide Pulleys. of the pulley toward which the belt is advancing shall be in line with the edge of the pulley that the belt is leaving, the same as in the case of the quarter turn belt shown in Fig. 198. It is not necessary that the shafts on which the main pulleys are mounted be in the same plane ; their direction may be such that their relation to 286 SELF-TAUGHT MECHANICAL DRAWING each other is similar to that of those shown in Fig. 198, or at any intermediate angle. Again, if they are in the same plane, it is not necessary that they should be parallel with each other; they may be at any angle with each other. Fig. 201 shows a case which is a modification of Fig. 200. The main shafts are at right angles to SHAFTS, BELTS AND PULLEYS 287 each other. The main pulleys, being of the same size, permit the guide pulleys to be mounted on a single shaft. This arrangement is a common method of transmitting power around a corner. Fig. 202 shows a case where the direction of the shafts with regard to each other is the same as in FIG. 202. A Case where the Guide Pulleys would be Mounted in an Adjustable Frame. Fig. 198, but where shop conditions are such that it is not practicable to bring the lower shaft under the upper one to permit of belting by either of the methods shown in Figs. 198 or 199. The guide pulleys are, therefore, mounted on a frame which can be raised or lowered in guides by means of an adjusting screw, permitting of an easy adjustment of the belt tension. 288 SELF-TAUGHT MECHANICAL DRAWING Fig. 203 shows a case which is similar to Fig. 200 in that it permits the belting together of shafts which are at angle to each other, but accomplishes this result by the use of only one guide pulley. The shafts, though at an angle to each other, are in the same plane. This, however, is not neces- sarily so. The shafts may be twisted around until they are at right angles to each other, as in Fig. 198. As shown in Fig. 200, the belt may be run in either direc- tion as long as the shafts are in the same plane; but as shown in Fig. 203, it is nec- essary that the belt should be run in the direction in- dicated by the arrows. An examination of the en- gravings will show that the condition necessary for the proper working of guide pulleys is that the shaft on which the guide pulley is mounted shall be at right angles to a line drawn from the edge of the pulley that the belt is leaving in its advance toward the guide pulley, to the middle of the guide pulley face. FIG. 203. An Arrange- ment in Which but One Guide Pulley is Used. CHAPTER XVIII FLY-WHEELS FOR PRESSES, PUNCHES, ETC. IN a great many different classes of machinery, the work that the machine performs is of a variable or intermittent nature, being done, in the case, for example, of punches and presses, during a small part of the time required for the driving shaft or spindle of the machine to make a complete revolu- tion. If this work could be distributed over the entire period of the revolution, a comparatively nar- row belt would be sufficient to drive the machine; but a very broad and heavy belt would otherwise be necessary to overcome the resistance, if the belt only be depended on to do the work. It is, of course, in a sense, impossible to distribute the work of the machine over the entire period of revolution of the driving shaft of the machine, but by placing a large, heavy-rimmed wheel, a fly-wheel, on the shaft, the belt is given an opportunity to perform an almost uniform amount of work during the whole revolution. During the greater part of the revolution of the driving shaft the power of the belt is devoted to accelerating the speed of the fly- wheel. During that brief period of the revolution of the shaft when the work of the machine is being done, the energy thus stored up in the fly-wheel is given out at the expense of its velocity. The 289 290 SELF-TAUGHT MECHANICAL DRAWING energy a fly-wheel would give out if brought to a standstill would be (neglecting the weight of the arms and hub, as the efficiency of the wheel depends chiefly on the weight of the rim), expressed in foot-pounds, equal to the weight of the rim in pounds multiplied by the square of its velocity at its mean diameter in feet per second, and this product divided by 64.32, the same as in the case of a falling body moving at the same velocity, as explained in the section on mechanics. Expressed as a formula this rule is : = Wv* Wv 2 2g 64.32 in which E = total energy of fly-wheel, W = weight of fly-wheel rim in pounds, v = velocity at mean radius of fly-wheel in feet per second, g = acceleration due to gravity = 32. 16. If the speed of the fly-wheel is only reduced, the energy which it would give out would be equal to the difference between the energy which it would give out if brought to a full stop, and that which it would still have stored up in it at its reduced velocity. Therefore, to find the energy in foot- pounds which a fly-wheel will give out with an allowable loss of speed, subtract the square of the velocity of the rim in feet per second at its reduced speed from the square of its velocity in feet per second at full speed, multiply this difference by the weight in pounds, and divide the product by 64.32. The result will give the loss of energy in foot-pounds. FLY-WHEELS 291 This long and cumbersome rule is expressed much more simply by the formula: ElS= ""64732" in which E l = energy, in foot-pounds, fly-wheel gives out while speed is reduced from Vi to v 2 , Vi = speed before any energy has been given out, in feet per second, v 2 = speed at end of period during which energy has been given out, in feet per second, W = weight of fly-wheel rim in pounds. This rule and formula may be transposed as fol- lows : To find the weight of a fly-wheel to give out a required amount of energy with an allowable loss of speed, multiply the required amount of energy in foot-pounds by 64.32, and divide the product by the difference between the square of the velocity of the rim, at its mean diameter, in feet per second at full speed, and the square of its velocity in feet per second at its reduced speed ; or, expressed as a formula, using the same notation as above : v i 2 v ? When the mean diameter of the fly-wheel is known, the velocity of the rim at its mean diameter in feet per second will be Diameter in feet X 3.1416 X rev, per minute ~60 It is evident that in designing a fly-wheel for a 292 SELF-TAUGHT MECHANICAL DRAWING machine, there is an opportunity for a wide range in the weight, from a wheel heavy enough, when once it has been brought to its full speed, to do, by means of the energy stored in it, the work without assistance from the belt, the belt being only just wide enough to restore the speed of the wheel in time for the next operation, to a wheel where the belt is wide enough to do the most of the work directly, the stored energy in the fly-wheel merely assisting it somewhat. Perhaps the best way would be to have the wheel heavy enough so that its stored energy could do the bulk of the work, the belt assisting it, and at the same time have the latter wide enough to quickly restore the speed of the wheel, so that, in case its velocity should be reduced beyond that calculated, there would be a margin of available power in the belt. Example. Let it be required to design a fly- wheel for a press to cut off one-inch round bar steel, the press making 30 strokes per minute. Soft steel having a shearing resistance of about 50,000 pounds per square inch, and a one-inch bar having an area of cross-section of 0.7854 square inch, the shearing resistance of the bar will be 50,000 X 0.7854 = 39,270 pounds, or practically 40,000 pounds. This resistance varies,, however, during the process of shearing, being greatest near the beginning of the cut, and decreasing as the cutting progresses. In the case of a round bar it could not decrease uniformly, because of the shape of the cross-section. For the sake of getting the decrease in resistance as nearly uniform as possible, we will assume that the work of cutting off a one- FLY-WHEELS 293 inch round bar is the same as the work of cutting off a square bar of the same area ; though this may not be quite exact, it would probably not be far out of the way. The length of the sides of a square of the same area as a given circle, is equal to the diameter of the circle multiplied by 0.886. There- fore, our equivalent square bar will be 0.886 of an inch square. The mean resistance to cutting, assuming that the resistance decreases uniformly as the cutting progresses, would be 40,000 ^ 2 = 20,000 pounds. As the cutting operation continues through a space of 0.886 of an inch, the power required would be 20,000 X 0.886 = 17,720 inch- pounds, or 1476.6 foot-pounds. Let us plan to have the belt do one-fifth of the work of cutting direct- ly, leaving four-fifths to be done by the stored up energy of the fly-wheel. One-fifth of 1476.6 equals 295.3. Subtracting this from 1476.6 leaves 1181.3 foot-pounds to be supplied by the energy of the fly-wheel. As a preliminary calculation let us find what would have to be the weight of the wheel if it were to be placed upon the crank-shaft, the shaft which operates the plunger of the press. Assuming the mean diameter of the fly-wheel rim to be 4 feet, the circumference would be 4 X 3.14 = 12.56 feet, and, as the shaft makes 30 revolutions per minute, the velocity of the rim in feet per second would be : 12.56 X 30 c 00 , gg = 6.28 feet. If we expect the fly-wheel to suffer a loss of, say, 10 per cent, while doing its work, then its velocity at its reduced speed will be 6.28 - 0.628 = 294 SELF-TAUGHT MECHANICAL DRAWING 5.65 feet. The weight of the fly-wheel to give out 1181. 3 foot-pounds under these conditions will then be, according to the rule and formula already given : 1181.3 X 64.32_ 75,981.2 75.981.2 _ 1 n 6.28 2 -5.65 2 39.44-31.92 = 7.52 nearly. A wheel weighing 10, 100 pounds would, of course, be out of the question ; but as the energy increases as the square of the velocity, the weight may be very rapidly reduced by mounting the wheel upon a higher-speeded secondary shaft, connected with the crank-shaft by reducing gears. If the speed of the secondary shaft is to the speed of the crank- shaft as 6 to 1, the weight of the wheel, if the mean diameter be kept the same, will need to be only about one thirty-sixth of what it would need to be if mounted on the crank-shaft. At thisjhigher speed, however, it might be desirable to somewhat reduce the diameter of the wheel. Let us assume that the mean diameter be made 3 feet. If the ratio of speeds is 6 to 1, the wheel will make 180 revolutions per minute, and the velocity of the rim in feet per second will be : 3 X 3.14 X 180 ' , To" =28-3 feet. If the wheel suffers a loss of 10 per cent., its velocity at its reduced speed will be 28.3 - 2.83 = 25.5 nearly. The weight of the wheel will then be: 1181.3 X 64.32 75,981.2 Kn A , 28.3 2 - 25.5 2 = 15064^ = 5 4 P Unds ' FLY-WHEELS 295 As a cubic inch of cast iron weighs 0.26 pound, the wheel will contain . 504 -5- 0.26 = 1938 cubic inches. The mean circumference of the rim in inches will be 3 X 12 X 3.14 = 113 inches. The cross-section of the rim will then be : 1938 - 113 = 17.1 square inches. This would mean a rim about 4 by 4J inches. The outside diameter of the wheel would then be 40 inches. We planned to have the belt do one-fifth of the work, and this we found to be 295.3 foot-pounds. If the crank has a radius of li inch, the cutter will have a stroke of 2J inches, and if the cutters over- lap each other one-quarter of an inch at the end of the stroke, the crank will have to swing through an angle of about 54 degrees in order to make the cutters advance the one inch necessary to cut off the one-inch bar, as a simple lay-out will show. The belt must then develop 295.3 foot-pounds while the crank swings through 54 degrees. It will then develop 295.3 + 54 = 5.5 foot-pounds, nearly, in one degree, and in a complete revolution it will develop 5.5 X 360 = 1980 foot-pounds. As the press makes 30 strokes per minute, the belt will develop 30 X 1980 = 59,400 foot-pounds per minute. If a driving pulley 18 inches in diameter is used, the belt speed in feet per minute will be : 18 X 3.14 X 180 a , ^ = 848 feet. If a single thickness belt, one-inch wide, at 1000 feet per minute, transmits 33,000 foot-pounds 296 SELF-TAUGHT MECHANICAL DRAWING per minute, the same belt at 848 feet per minute will transmit T Wo as much, or 33,000X0.848 = 27,984 foot-pounds. The width of belt necessary to transmit 59,400 foot-pounds per minute at this speed will then be 59,400 -*- 27,984 = 2.1 inches. No account has so far been taken of the power necessary to drive the machine itself. To allow for this the belt should evidently be not less than 2i inches wide. A 3-inch belt would allow consid- erable of a margin of safety, and further calculation will show that such a belt would develop, during about one- third of a revolution of the crank, the amount of energy which the fly-wheel had lost, so that, as the cutting operation takes about one-sixth of a revolution, the fly-wheel would be running at full speed for about one-half of a revolution of the crank, previous to the beginning of the cut, pro- vided that it had not suffered any greater reduction of velocity than the 10 per cent, planned for. If the press was employed doing punching the same method of procedure would be employed in the calculations, the area in shear in such a case being equal to the circumference of the hole mul- tiplied by the thickness of the plate. The end of a punch . is usually made slightly conical or slightly beveling, the effect in either case being to increase the shearing action, and make the work of punch- ing easier. CHAPTER XIX TRAINS OF MECHANISM FOR obtaining high speeds without the use of unduly large driving pulleys or gears, for securing gain in power by sacrificing speed, for securing reversal of direction, or for obtaining some par- ticular velocity ratio between the driver and some part of the mechanism, pulleys, gears, worm-gears, or the like, may be substituted for direct acting driving-mechanisms. To Secure Increase of Speed. Let a shaft making 100 revolutions per minute be required to drive the spindle of a machine at 2000 revolutions per minute, the pulley on the spindle being 3 inches in diam- eter. If a direct drive were to be used, the pulley on the shaft would have to be as many times greater than the pulley on the spindle as 2000 is greater than 100, or 20 times. This would mean a pulley on the shaft 60 inches in diameter. Practical considerations, such as the weight of the pulley, size of hangers and the like, would make such a pulley out of the question. By interposing an intermediate countershaft be- tween the first shaft and the spindle of the machine, however, having pulleys of such size that the product of the ratio of the pulley on the first shaft and the one to which it is belted on the counter- shaft, multiplied by the ratio of the second pulley 297 298 SELF-TAUGHT MECHANICAL DRAWING on the countershaft and the pulley on the spindle to which it is belted is equal to the ratio which it is desired to have between the first shaft and the spindle, the same speed may be secured by the use of pulleys of convenient size. Thus, if the ratio between the pulley on the first shaft and the one on the countershaft is as 1 to 4, and the ratio between the driving pulley on the countershaft and the one on the spindle of the machine is as 1 to 5, the product of these two ratios, 1 to 4 and 1 to 5, is 1 to 20, and the arrangement will give the FIG. 204. Reversal of Direction Obtained by Crossed Belt. required speed. The pulley on the spindle being 3 inches in diameter, the driving pulley on the coun- tershaft will be 15 inches in diameter, and if the driven pulley on the countershaft is 4 inches in diameter the pulley on the first shaft to which it is belted will be 16 inches in diameter, instead of 60 inches, as would be required with direct belting. If the spindle of the machine, instead of being driven were made the driver, as it would be if it were the armature shaft of a motor, then this ar- rangement would give gain in power with con- sequent loss of speed. To Secure Reversal of Direction. In cases where shafts are belted together, reversal of direction of TRAINS OF MECHANISM 299 rotation is secured by simply using a crossed belt instead of an open one, as shown in Fig. 204. When gears are used, reversal of direction of rota- tion follows as a natural condition of their meshing together, as shown in Fig. 205. In order that the two gears A and B shall rotate in the same direc- tion, it is necessary to separate them slightly, and interpose an intermediate gear, or idler, between FIG. 205. Relative Direc- tion of Rotation in a Pair of Gears. FlG. 206. Influence of Idler on Direction of Rotation. them as shown in Fig. 206. The rates of rotation of A and B with regard to each other is not affected by the idler gear, whether the idler be large or small. That this is so may be seen by direct exam- ination. If A is the driver, its circumference will impart to the circumference of C its own rate of motion, and C will in turn impart to B the same rate of motion, which is the same as it would have if in direct connection with A. If, now, another idler be interposed between A and B, making four gears in the train, A and B will again rotate in opposite directions. From this it will be seen that when a train is composed of an 300 SELF-TAUGHT MECHANICAL DRAWING even number of gears, the first and last members rotate in opposite directions ; but when the train is composed of an odd number of gears, the first and last members rotate in the same direction. In Fig. 207 is shown the mechanism used in engine lathes to secure either direct or reversed motion, by having the working train consist of either an even or an odd number of gears. In this FIG. 207. Principle of Turn- FIG. 208. Principle of Com- bler Gear. pound Idler. arrangement A is a gear on the head-stock spindle, and B is a gear on a stud below. Pivoted on the axis of B is a triangular piece of metal, or bracket, shown in dotted lines, which can be swung back and forth by the handle E. Mounted on this bracket are the idler gears C and D, C being con- stantly in mesh with B, and D being in mesh with C. When it is required that B shall rotate in the same direction as A, the handle E is lowered until C meshes with A. The working train then consists TRAINS OF MECHANISM 301 of three gears, A, C and B, D being out of mesh with A, revolving by itself, but not forming a part of the working train. When it is desired that B shall rotate in the opposite direction to A, the handle E is raised until D meshes with A, C being thrown out of mesh with it. The working train then con- sists of four gears, A, D, C and B, and the desired reversal is secured. The Compound Idler. It has been shown that when a train consists of simple gears the relative rates of rotation of the first and last members re- main unchanged, regardless of the number or size of the idlers that may be interposed. When it is desired to secure a different rate of rotation be- tween two members of a train than that which they would have if meshing directly together, a compound idler is used, as shown in Fig. 208. Such a gear is used on many screw cutting lathes. For cutting threads up to a certain number per inch the screw cutting train consists of simple gears. A compound idler may then be introduced into the train, when without other change additional threads may be cut. If with screw cutting trains of simple gears a lathe will cut all whole numbers of threads up to 13 threads per inch, then, by adding a compound idler to the train, having its two steps in the ratio of 2 to 1, threads from 14 to 26 per inch (except odd numbers) may be cut with the same gears as previously used for cutting up to 13 threads per inch. If the compound idler forms an additional member of the train, the reversal of direction of rotation which would take place in the motion of the lead-screw of the lathe may be taken care of 302 SELF-TAUGHT MECHANICAL DRAWING by the reversing gears between the spindle of the head-stock and the stud, previously described, and shown in Fig. 207. The Screw Cutting Train. In Fig. 209 is shown the screw cutting mechanism found on engine lathes. The reversing mechanism shown in Fig. FIG. 209. FIG. 210. FIGS. 209 and 210. Arrangement of Lathe Change Gearing. 207 is reproduced entire, and these gears the gear A on the lathe spindle, the gear B on the stud, which is connected with A by the idlers C and D are all permanent gears. These gears are usually on the inside of the head-stock as shown in Fig. 210. The stud reaches through the head-stock, and on its outer end is the change gear F, connecting with the change gear G on the lead-screw of the lathe by means of the intermediate idler H. The idler H is mounted on a slotted swinging arm as shown, so as to allow of gears F and G being TRAINS OP MECHANISM 303 replaced by others of such size as may be required to cut the particular screw desired. The carriage of the lathe, carrying the screw cutting tool, is driven directly by the lead-screw. On large lathes this screw is quite coarse, four threads per inch being common, while on smaller lathes a finer thread is used. The gear A on the spindle and the fixed gear B on the stud are sometimes of the same size, and sometimes of different sizes. The problem met with in screw cutting is to find what sizes change gears, F and G, must be used so that the lead-screw shall drive the carriage along one inch while the spindle of the lathe is making a number of revolutions equal to the number of threads to be cut per inch. Let us take as an example the assumed case of a lathe in which the lead-screw has 9 threads per inch, and in which the number of teeth in the gear on the spindle is to the number of teeth in the fixed gear on the stud as 3 to 4; required the size of change gears to cut 23 threads per inch. Then, as the lead-screw has 9 threads per inch, the spindle of the lathe must make 23 revolutions while the lead-screw is making 9 revolutions. The method used in a previous chapter for obtaining the size of pulleys to give required speeds will give us the solution of this problem; if the speed of the first driving member of the train, together with the number of teeth or relative sizes of all other driving members be placed on one side of a vertical line, and the speed of the last driven member, together with the number of teeth or relative sizes of all other driven members be placed on the other side of the line, the product 304 SELF-TAUGHT MECHANICAL DRAWING of the numbers on one side of the line multiplied together will equal the product of the numbers on the other side of the line multiplied together. The spindle of the lathe is, of course, the first driving member of the train, and the lead-screw is the last driven member. As the spindle is to make 23 revolutions while the lead-screw makes 9 revolu- tions, 23 will be the first number on the side of the line on which the driving members are placed, and 9 will be the last number on the side of the line on which the driven members are placed. Next, as the ratio between the sizes of the driving gear on the lathe spindle and the fixed gear on the stud below which it drives is as 3 to 4, these numbers will be placed against each other on opposite sides of the line. The ratio between the numbers of teeth or sizes of the two change gears, F and G, whose sizes it is required to find, being unknown, may be said to be as 1 to the unknown number x. These numbers, 1 and 05, are now placed on their proper sides of the line, and the problem appears as shown below. The size of the idler gear H does not enter into the question, because, as has been previously shown, a simple idler gear does not affect the relative rates of rotation of the gears between which it transmits motion. Speed of spindle 23 Ratio of size of spindle gear 3 4 to size of fixed stud gear. Ratio of number of teeth in change gear F 1 ! x to number of teeth in change gear G 9 speed of lead-screw. 69 = ZGx TRAINS OF MECHANISM 305 Multiplying together the numbers on both sides of the line gives the equation 69 = 36x. It is evi- dent that if 69 equals 36x, x must be equal to 69 divided by 36, or f|. The ratio between sizes of the gear F and the gear G is then as 1 to ||. Eliminating the fraction by multiplying both terms of the ratio by 36 gives the ratio as 36 to 69. If, then, F has 36 teeth, and G has 69 teeth, the lathe will cut the required number of 23 threads per inch. In Fig. 211 is shown how a compound idler gear is sometimes used in a screw cutting train. The FIG. 211. Compound Gearing. change gear G and the idler H have long hubs on one side. When it is desired to cut finer threads than what the gears E and G with the idler H will give, H and G, are put on with the long hubs toward the lathe, throwing them out of line with E. The gear E then meshes into the large step of 7, the small step of / meshes into H, and H meshes 306 SELF-TAUGHT MECHANICAL DRAWING into G. The ratio between the large and the small steps of / must then be taken into account in the calculation. For cutting the coarser threads H and G are put on with the short hubs toward the lathe, bringing them into line with E. The idler / is also turned over, so that its large step is on the outside and out of line with E and H. It is then swung back out of the way. When the gearing is fully compounded the two gears at / are separate from each other but keyed together on the same stud and mounted in the same manner as shown in Fig. 211. By varying the sizes of these gears, almost any screw thread may be cut within reasonable limits. In this case, of course, there are four gears to be determined in our calculations. Simplified rules are given in the following for this case, as well as for the regular simple trains. Large lathes are provided with change gears for cutting threads from about 2 to about 20 threads per inch, smaller lathes being provided with gears for cutting from about 3 or 4 to 40 or 50 threads per inch, in either case including a pair of gears for cutting 11J threads per inch, this being the stand- ard thread for iron pipes from one to two-inch sizes inclusive. The smaller lathes would also naturally be provided with gears for cutting 27 threads per inch, this being the number of threads on i-inch iron pipes. Simplified Rules for Calculating Lathe Change Gears. The following rules for calculating change gears for the lathe have been published by Ma- chinery (Reference Series Book No. 35, Tables TRAINS OF MECHANISM 307 and Formulas for Shop and Draftingroom) , and are here given because of their concise form and simplicity. Rule 1. To find the "screw-cutting constant*' of a lathe, place equal gears on spindle stud and lead- screw; then cut a thread on a piece of work in the lathe. The number of threads cut with equal gears is called the " screw-cutting constant " of that particular lathe. Rule 2. To find the change gears used in simple gearing, when the screw-cutting constant as found by Rule 1, and the number of threads per inch to be cut are given, place the screw-cutting constant of the lathe as numerator and the number of threads per inch to be cut as denominator in a fraction, and multiply numerator and denominator by the same number until a new fraction is obtained represent- ing suitable numbers of teeth for the change gears. In the new fraction, the numerator represents the number of teeth in the gear on the spindle stud, and the denominator, the number of teeth in the gear on the lead-screw. Rule 3. To find the change gears used in com- pound gearing, place the screw-cutting constant as found from Rule 1 as numerator, and the number of threads per inch to be cut as denominator in a fraction ; divide up both numerator and denomi- nator in two factors each, and multiply each pair of factors (one factor in the numerator and one in the denominator making a pair) by the same num- ber, until new fractions are obtained, representing suitable numbers of teeth for the change gears. The gears represented by the numbers in the new 308 SELF-TAUGHT MECHANICAL DRAWING numerators are driving gears, and those in the denominators driven gears. Two examples, showing the application of these rules, will be given in the following. Example 1. Assume that 20 threads per inch are to be cut in a lathe having a "screw-cutting con- stant," as found by the method explained in Rule 1, equal to 8. The numbers of teeth in the avail- able change gears for this lathe are 28, 32, 36, 40, 44, etc., increasing by 4 up to 96. By applying Rule 2, we have then : S_ _8_X_4 = 32 20 == 20 X 4 80 By multiplying both numerator and denominator by 4 we obtain two available gears having 32 and 80 teeth. The 32-tooth gear goes on the spindle stud and the 80-tooth gear on the lead-screw. It will be seen that if we had multiplied by 3 or by 5 instead of by 4, we would not have obtained avail- able gears in both numerator and denominator, as 8X3 would have given 24 and 20 X 5 would have given 100, both of which gears are not in our given set of gears. The proper number by which to multiply can be found by trial only. Example 2. Assume that 27 threads per inch are to be cut on the same lathe as assumed in Example 1. In this case the calculation must be made for compound gearing, as so fine a pitch could not be cut by simple gearing in this lathe. By applying Rule 3 we have : _8_ _2_>^4 (2 X 20) X (4 X 8) = 40 X 32 27 : = 3 X 9 "(3~X 20) X (9 X 8) = 60 X 72 TRAINS OF MECHANISM 309 The four numbers in the last fraction give the numbers of teeth in the required gears. The gears in the numerator (40 and 32) are the driving gears, and those in the denominator (60 and 72) are the driven gears. It makes no difference which one of the driving gears is placed on the spindle stud or which one of the driven gears is placed on the lead-screw. Back-Gears. Nearly all engine lathes and many other machine tools are provided with a set of re- c = FIG. 212. Principle of Back -Gearing. ducing gears, called back-gears, by means of which double the range of speeds that can be obtained by direct driving may be given to the spindle of the machine. Fig. 212 illustrates such a set of gears, and the method of applying them to the machine. The large gear A is fastened to the spindle of the machine, but the cone pulley, with the gear B attached to it, is loose on the spindle. The back- 310 SELF-TAUGHT MECHANICAL DRAWING gear shaft with gears C and D is mounted in brackets on the back side of the head-stock, and is provided with eccentric bearings, by means of which the gears on it can be thrown into or out of mesh with the gears on the head-stock spindle. When direct driving is desired, the back-gears are thrown back, out of the way, and the cone pulley and the large gear are clamped together by means of a screw pin or stud passing through the gear into the cone. They then revolve together as one piece. Let us assume the case of a lathe having a cone with four steps, the largest step being 6 inches in diameter, and the smallest 4 inches in diameter, with the intermediate steps in proper proportion. If the cone pulley on the countershaft is of the same size as the one on the spindle, then, if the countershaft runs 300 revolutions per minute, direct driving will give about the following speeds to the spindle: 450, 345, 260 and 200. Let it now be required to find the sizes of gears to be used so that with the back-gear driving, a proportionately slower rate of speeds may be obtained. We may solve the problem by giving to the gears some arbitrary sizes, and finding what speeds such sizes will give, and then modify these sizes until the required speeds are obtained. For trial purposes let us make the pitch diameter of the gear A the same as the diameter of the large step of the cone pulley, or 6 inches, and the pitch diameter of the gear B the same as the diameter of the small step of the cone pulley, or 4 inches. Arranging driving and driven members on opposite sides of a vertical TRAINS OF MECHANISM 311 line, the speed of the first driving member of the train, the countershaft, being 300, the required speed of the last member, the lathe spindle, being represented by x, and having the belt on the largest step of the countershaft cone so as to obtain the highest speed with back-gears, gives an arrange- ment of the case as below. The sizes of the back- gears are the same as those on the lathe spindle, the gear C being 6 inches in pitch diameter, and the gear D 4 inches in pitch diameter. Speed of countershaft 300 Pulley on countershaft 6 Gear B on lathe 4 Back-gear D 4 4 Pulley on lathe 6 Back -gear C 6 Gear A on lathe x Speed of spindle 28,800= 144s From this it is seen that with the sizes of the gears as above, the highest speed with back-gears would be the same as the lowest speed without the back-gears. This, of course, would be useless duplication of speeds. For another trial we. will make the sizes of the gears B and D each 3J inches in pitch diameter. The calculation then becomes: Speed of countershaft 300 Pulley on countershaft 6 Gear B on lathe 3.5 Back-gear D 3,5 4 Pulley on lathe 6 Back-gear C 6 Gear A on lathe x Speed of spindle ) nearly. 312 SELF-TAUGHT MECHANICAL DRAWING A speed of 153 revolutions per minute for the fastest back-gear speed follows quite regularly the series of speeds which the direct drive gives. Instead of using the pitch diameters of the gears in making the calculations the number of teeth which the gears would have, the pitch being first decided on, might be used. In this manner it is possible to make slight changes in the diameters of the gears without bringing troublesome fractions into the calculations. Many lathes and other machine tools have trains of mechanism much more complicated than any here shown, but the method of procedure here outlined can be applied to all of them. CHAPTER XX QUICK RETURN MOTIONS IN a large class of machinery the work is done during the forward motion of a reciprocating part; the return of the part to its starting point is then a question of time. The quicker the part can be returned to its starting point, the more efficient becomes the machine. When the stroke is long, as in the case of the bed of an iron planer for large work, this rapid return motion is usually obtained by means of shifting the driving belt onto a return pulley so arranged that a higher ratio of speed is procured; but in other cases, where the recipro- cating motion is shorter, and the stroke is actuated by means of a crank, the actuating mechanism is made such that the crank gives a slow forward and a quick return motion to the reciprocating part. Iron planers for small work, shapers, and the like, and some classes of engines and pumps, use such quick return motions. Below are described the principal devices used for such purposes. Fig. 213 shows a method of securing a quick return by having the axis of the crank outside of the path of the reciprocating end of the connecting- rod. Let A be a crank, the crank-pin of which, a, acting upon the connecting-rod B represented by the heavy line, causes the block b to move back and 313 314 SELF-TAUGHT MECHANICAL DRAWING forth in the path CD. When the crank is in the position shown the block is at the extreme left of its stroke, the connecting-rod and crank being in the same straight line, the center line of the con- necting-rod coinciding with the axis of the crank. As the crank swings downward, the block b is driven to the right; but an examination . of the illustration will show that the crank must make FIG. 213. Simple Quick Return Motion. more than a half revolution before it again forms a straight line with the connecting-rod, which it will do when the block has reached its extreme position to the right. As, therefore, the block makes its movement to the right while the crank is swinging through the lower angle included be- tween these two positions, and as it makes its return stroke while the crank is swinging through the upper angle included between these same two positions, the time of the forward stroke of the block -will be to the time of its return stroke as the lower angle is to the upper angle. QUICK RETURN MOTIONS 315 The upper angle being the smaller of the two, the block has a quick return motion. To secure ease of motion to the block as it starts on its stroke to the right, the angle abC, the angle which the connecting-rod makes with the path of the block, should not be more than about 45 degrees. To design a quick return motion of this type, lay out a horizontal line ab, Fig. 214, and on it mark off cb equal to the required length of stroke. From c draw the line cd of indefinite length at such an fl b FIG. 214. Lay-out of Quick Return Motion in Fig. 213. obliquity that the angle acd shall not be more than 45 degrees. From b draw the line be at the angle required to give the desired quick return. The intersection of these two lines at /will be the axis of the crank. The length bf will be seen by re- ferring back to Fig. 213 to be equal to the length of the crank plus the length of the connecting-rod. The length of cf will be seen to be equal to the length of the connecting-rod minus the length of the crank. If in a given case the length cb is made 12 inches, and cf is found to be 10 and bf 21 inches, which they would be if the angles were as 316 " SELF-TAUGHT MECHANICAL DRAWING shown in Fig. 214, then, letting x represent the length of the connecting-rod and y the length of the crank, we would have x + y = 21 inches, and x - y = 10 inches. Adding the left-hand and the right-hand members, respectively, of these two equations, we would have x + y + x-y = 21 + 10 = 31 inches. As + y - y= we may eliminate these expressions, and the equation will read 2x = 31 inches, and x, the length of the connecting-rod, will thus be 15J inches. The length of the crank will then be 21 inches (the length of bf) minus 15J inches, or 5J inches. It will be seen that if the length of the stroke is made variable by having the crank-pin, a, adjust- able to different positions on the crank A, Fig. 213, the difference between the time of the forward and of the return stroke of the sliding block b will be lessened, because the two positions which it will occupy at. the extremes of its stroke will be nearer together, and the lower and upper angles which the crank passes through in giving to the block its forward and return movements will be more nearly equal. Fig. 215 shows a quick return motion device especially adapted to cases where the horizontal space is limited, and which is much used on shapers. The illustration shows a shaper in outline. The ram of the shaper is given its forward and return motion by means of the rocking arm A, which swings on a fulcrum at B. The rocking arm is given its motion by means of a crank-pin on the disk C, the pin engaging in a sliding block which travels in a slot in the arm A. QUICK RETURN MOTIONS 317 Let BC and BD, Fig. 216, represent the extreme positions of the rocker arm A. Draw the lines OF and OG from the center of the crank disk at O at right angles to BC and BD. It is evident that in order that the crank, on its upper sweep, shall FIG. 215. Diagram of Quick Return Arrangement in a Shaper. move the rocker arm from C to Z), it must move through the arc FAG, while to return the arm from D to C, on its lower sweep, it must move only through the lower arc FG. The time of the return motion will therefore be to the time of the forward motion as the lower arc or angle FG is to the arc 318 SELF-TAUGHT MECHANICAL DRAWING or angle FAG. If the crank is shortened so as to give a shorter stroke to- the ram of the shaper, then the rocker arm will swing through a smaller angle, as from H to /, and lines drawn from at FIG. 216. Diagram of Speed Ratios in Shaper Motion. right angles to HB and IB will be more nearly in a straight line than OF and OG. There will, there- fore, be less difference between the time of forward and return motions on short strokes than on long ones. QUICK RETURN MOTIONS 319 The Whitworth Quick Return Device. Let A, Fig. 217, be a slotted arm revolving on its axis at B. Above A is the driving crank C, having a pin engaging in the slot at the left in the arm A. The slot at the right in the arm A is provided for an adjustable stud which drives the reciprocating parts, through the medium of the connecting-rod _D FIG. 217. -Whitworth Quick Return Motion. D. It will be seen that, as shown, the connecting- rod is at the extreme right of its motion, forming as it does a straight line with the revolving arm A, which latter is at the same time at right angles with the center line cd. It will be seen that in order that the arm A may move through half a revolution so as to bring the connecting-rod to the extreme left of its motion, it will be necessary for the actuating crank C to revolve either through the 320 SELF-TAUGHT MECHANICAL DRAWING upper angle x or through the lower angle y, so as to form again the same angle with the center line cd, but at the right of it, as it is now shown form- ing with it at the left. The forward and return motions will, therefore, be to each other as the angle x is to the angle y. To design a quick return motion of this type it is, therefore, necessary to first lay out the angles x and y of such relative sizes that x is as many times greater than y as the time of the forward motion is to be greater than the time of the return motion, having them, of course, central on the line cd. The distance apart of the f ulcrums of the crank C and of the revolving arm A will be partly determined by the sizes of their shafts. The location of the crank-pin, de- termining the length of the crank, will then be at the intersection of the horizontal center line of the revolving arm A with the dividing line ef between the angles x and y. The length of the crank must, of course, be sufficient so that the crank pin will swing under the hub of the arm A, and the length of the crank-pin slot in A must be. sufficient for the motion of the pin relative to the arm. It will be noticed that, unlike the two preceding quick return devices, varying the stroke of the reciprocating parts does not alter the relative time of the forward and return motions ; for such change does not affect the angles x and y upon which the time of the forward and return motions depends. If, however, the length of the crank C is varied, then the angles x and y are altered, and the time of the forward and return motions will be affected. It will be seen upon examination that with the QUICK RETURN MOTIONS 321 construction shown the revolving arm A must be made in two parts, one at each end of its shaft, in order to avoid interference of the parts of the mechanism with one another as they revolve. This trouble is overcome by replacing the crank C with a crank disk which fits over and revolves upon a fixed stud or hub large enough to receive the stud at B upon which the arm A revolves. The Elliptic Gear Quick Return. If two ellipses of equal size, Fig. 218, having foci at w and x and FIG. 218. Quick Return Motion by Means of Elliptic Gears. at y and z, be placed in contact with each other with their long diameters forming a continuous straight line as shown; then if the ellipses are caused to, revolve freely upon their correspond- ing foci, w and y, they will roll upon each other perfectly, without slipping. From the nature of an ellipse as shown by its construction with a thread and pencil (see Chapter III, Problem 13) it will be seen that if the ellipse at the left were being formed in this manner and the pencil were at D, the intersection of the circumference of the 322 SELF-TAUGHT MECHANICAL DRAWING ellipse with the long diameter, the length of the thread would be equal to the sum of the distances wD and Dx. But the distance Dx is the same as the distance Dy\ therefore, the length of the thread would be equal to the distance wy, the distance between the foci upon which the ellipses are re- volving. If, now, the ellipses are revolved until the points A and B, vertically over the foci x and y, are in contact with each other, the sum of the distances wA and By will be equal to the distance between the foci w and y, for their sum is equal to the length of the thread, and the length of the thread is equal to wA plus Ax, and Ax is equal to By, as points A and B are both vertically over the foci of the ellipses. In a similar manner any pair of points may be selected on the two ellipses equally distant from the point D. The distance from the point on the ellipse at the left, to the focus w, will be equal to the length of the thread at the left of the pencil, and the distance from the point on the ellipse at the right, to the focus y, will be equal to the length of the thread at the right of the pencil, and their sum will be equal to the distance between the foci w and y. This distance between the foci w and y will be seen on further examination to be equal to the long axis of the ellipse. This property of the ellipse has been taken advantage of to secure a quick return motion to a reciprocating part of a machine. If in Fig. 218 the two ellipses represent the pitch lines of elliptic gears; with the gear at the left as the driver with a uniform motion, the one at the right will have an ununiform motion. If, now, a crank is mounted on the same shaft as QUICK RETURN MOTIONS 323 the driven elliptic gear, the crank having its center line at right angles to the long axis of the ellipse, and this crank actuates a sliding block back and forth in the direction of the center line of the two gears, then this block will have a slow motion in one direction, and a quick motion in the other direction. If, now, the gears are revolved from the position in which they are shown until A and B are in contact, the gear at the right will have made a quarter of a revolution and the sliding block will be at the extreme right of its stroke; but while this gear has made a quarter of a revolution, the driving gear has revolved through the angle AwD only. If, now, the gear at the right is revolved another quarter of a turn, the points E and F will be in contact, and the crank will be directed ver- tically upward. The driving gear will, however, have revolved through the angle AwF. The forward and return motions of the sliding block will, there- fore, be to each other as the angle AwF is to the angle AwD. In designing a pair of elliptic gears, therefore, the first thing to do is to determine the size of the angle Awx. To find the distance be- tween the foci w and x first lay out on a large scale a triangle similar to the triangle Awx. Then the sum of its hypothenuse and the perpendicular will be to the length of its base as the sum of wA and Ax (the long axis of the ellipse) is to wx, the dis- tance between the foci of the ellipse. The length of the short axis may then be found by reversing Problem 13, Chapter III. The problem may be solved even more accurately by the rules given for the solution of right-angled triangles. The length 324 SELF-TAUGHT MECHANICAL DRAWING of wA will be to Ax as 1 is to the sine of the angle Awx. Dividing the long axis of the ellipse into two parts in this proportion gives the length of wA and Ax. The length of wx will then be equal to the length of Aw multiplied by the cosine of the angle Awx. Then to find the short axis of the ellipse, divide the distance wx into two equal parts and construct the triangle wgh. The length wh will be half of the distance between the foci, and the length of wg will be half of the long axis. The length gh, half of the short axis, may then be found. Calculations made in this manner give the follow- ing proportions to ellipses for quick return ratios as indicated in the first column : Ratio of Forward to Return Motion. Long Axis. Short Axis. Distance Between Foci. 2 to 1 1.000 0.963 0.268 2* to 1 1.000 0.936 0.351 3 to 1 1.000 0.910 0.414 4 to 1 1.000 0.860 0.509 5 to 1 1.000 0.817 0.577 There appear to be two difficulties with elliptic gearing. The first is that if a high quick return ratio is attempted, so as to make considerable dif- ference between the long and the short axes, the obliquity of the action of the teeth upon each other, and the consequent great amount of friction between the teeth as they come together, becomes so great as to be troublesome. This may, to a con- siderable extent at least, be overcome by using a train of gears, each gear but slightly elliptic, in place of one pair of decidedly elliptic form. Thus QUICK RETURN MOTIONS 325 a train of three gears having their long and short axes in the proportion required to give a quick return of 3 to 1, with one pair of gears, will give a quick return of 9 to 1. If three gears of the 4 to 1 proportion are usad, a quick return of 16 to 1 will result. The second difficulty is that of correctly cutting the teeth. To work properly, the teeth should be cut on a machine having a special elliptic gear cutting attachment, otherwise the gears are likely to be expensive and unsatisfactory. Such an ellip- tical gear cutting arrangement is described, and the subject of elliptic gearing is quite fully dis- cussed, in Grant's treatise on gearing. Not being within the territory of this elementary treatise on machine design, the subject cannot here be dealt with in detail. INDEX Accelerated motion cams, 176 Acceleration of falling bodies, 143 Acme standard screw thread, 253 Addendum of gear teeth, 193 Aluminum, strength of, 162 Angle, definition of, 10 Angle of cone clutches, 271 Angle, to bisect an, 17 Angles, laying out, 118 Areas of plane figures, 92 A. S. M. E. standard machine screws, 258 Assembly drawings, 52 B Back gears, 309 Beams, cross-sections of, 156 Beams, strength of, 159 Belt for reversal of motion, crossed, 298 Belting, horse-power of, 277 Belting, speed of, 279 Belting, twisted and unusual cases of, 282 Belts, 276 Belts, endless, 278 Belts, laced, 278 Belts, width and thickness of, 277 Bending, shape of parts to resist, 155 Bending strength of beams, 159 Bevel gearing, calculating, 230 Bevel gears, 202 Blue printing, 78 Bolt heads, table of United States standard, 246 Bolts, studs and screws, 243 Bolts to withstand shock, 248 Brass, strength of cast, 162 Brass wire, strength of, 158 Broken drawings of long ob- jects, 73 Cam curve for harmonic mo- tion, 181 Cams, comparison between uniform motion and accele- rated motion, 183 Cams for high velocities, 175 Cams, general principles, 164 Cams with grooved edge, 172 Cams with pivoted follower, 167 Cams with positive return, double, 173 Cams with reciprocating mo- tion, 171 Cams with roller follower, 168 Cams with straight follower, 165 Cams with uniform motion, 165 Cams with uniformly accele- rated motion, 176 Cap screw sizes, 248 327 328 INDEX Case for drawing instru- ments, 4 Cast iron, strength of, 157 Castings, stresses in, 162 Change gears, for screw cut- ting, 302 Check or lock nuts, 248 Chord of circle, definition of, 12 Circle, area and circumfer- ence of, 92 Circle, area of, 83 Circle, circumference of, 80 Circle, definition of, 11 Circle, to find center of a, 19 Circles, circumscribed and in- scribed, 20 Circles, concentric, 10 Circles in isometric projec- tion, 48 Circular pitch, 205 Circular sector, area of, 93 Circular segment, area of, 93 Clamp coupling, 262 Clutches, friction cone, 269 Clutches, friction disk, 266 Clutches, toothed, 265 Compasses, 3 Complement angle, definition of, 11 Composition of forces, 120 Compound idler gear, 301 Compound gearing for screw cutting, 305 Compression of machine parts, 154 Compressive strength of ma- terials, 158 Concentric circles, 10 Cone and cylinder intersect- ing, 44 Cone clutches, angle of, 271 Cone clutches, friction, 269 Cone pulleys, 239 Cone pulleys, method of lay- ing out, 242 Cone, surface development of a, 40 Copper, strength of cast, 162 Cosecant of an angle, 102 Cosine of an angle, 101 Cosines, table of, 105 Cotangent of an angle, 102 Cotangents, table of, 107 Coupling, Hooke's, 263 Couplings, 259 Couplings, clamp, 262 Couplings, flange, 260 Crank motion, quick return, 313 Cross-sectioning device, 7 Cross-sections of beams, 156 Cube, projections of a, 39 Cube root, 82 Cube, volume of, 94 Cutting screw threads, gear- ing for, 302 Cylinder and cone, intersect- ing, 44 Cylinder, volume of, 94 Cylinders, intersecting, 43 Cycloid, definition of, 15 Cycloid, to draw a, 27 Cycloidal gear teeth, approx- imate shape of, 209 D Dedendum of gear teeth, 193 Definitions of terms, 10 Degree, definition of, 96 Detail drawings, 53 Diametral pitch, 207 Differential pulleys, 134 Disk clutches, friction, 266 Dimensions on drawings, 56 Double cams with positive return, 173 Drawings, assembly, 52 Drawing board, 1 Drawings, classes of lines on, 55 Drawings, detail, 53 Drawings, dimensions on, 56 Drawing instruments, 1 Drawing paper, 8 INDEX 329 Drawing pens, the use of, 7 Drawings, sectional views on, 66 Drawings, working, 50 E Efficiency of screws, 253 Elevation, definition of, 33 Ellipse, area of, 95 Ellipse, definition of, 14 Ellipse, to draw an, 21 Elliptic gear quick return motion, 321 Elliptic gear return motion, table for lay-out of, 324 Energy and work, 146 Energy of fly-wheel, 290 Engines, horse-power of steam, 81 Epicycloid, definition of, 15 Epicycloidal gearing, 191 Epicycloidal and involute systems of gears, compari- son between, 199 Erasing shield, 9 Factor of safety, 151 Falling bodies, 142 Finishing marks on drawings, 63 Flange couplings, 260 Foot-pound, definition of, 146 Force of a blow, 147 Forces, oblique, 124 Forces, opposing, 125 Forces, parallel, 123 Forces, resultant of, 120 Forces, resolution of, 123 Formulas, algebraic, 79 Formulas, transposition of, 88 Friction cone clutch, horse- power of, 270 Friction cone clutches, 269 Friction disk clutch, horse- power of, 267 Friction disk clutches, 266 Fulcrum, definition of, 126 Fly-wheel, energy of, 290 Fly-wheels for presses, punches, etc., 289 Fly-wheel, weight of, 291 Gear, compound idler, 301 Gear, influence of the idler, 299 Gear quick return motion, elliptic, 321 Gear teeth, approximate shape of, 209 Gear teeth, laying out invo- lute, 210 Gear teeth, Lewis' formula for strength of, 218 Gear teeth, pitch of, 205 Gear teeth, proportions of, 207 Gear teeth, strength of, 213 Gear teeth systems, compari- son between, 199 Gear tooth, hunting, 209 Gear tooth terms, definitions of, 193 Gear, tumbler, 300 Gearing, back, 309 Gearing, calculating bevel, 230 Gearing, calculating dimen- sions of, 222 Gearing, calculating spur, 222 Gearing, calculating worm, 234 Gearing, epicycloidal, 191 Gearing for reversal of direc- tion of motion, 299 Gearing for screw cutting, 302 Gearing, general principles of, 190 Gearing, worm, 204 Gears, bevel, 202 330 INDEX Gears, interference in in- volute, 198 Gears, involute, 196 Gears, knuckle, 190 Gears, method of drawing, 68 Gears, proportions of, 213 Gears, shrouded, 201 Gears, speed ratio of, 220 Gears, twenty degree invo- lute, 201 Gears with radial flanks, 195 Gears with strengthened flanks, 195 Geometrical problems, 17 Grooved edge cams, 172 Guide pulleys for belts, 285 Instrument case, 4 Involute and epicycloidal sys- tems of gears, comparison between, 199 Involute, definition of, 15 Involute gears, 196 Involute gears, interference . in, 198 Involute gear teeth, laying out, 210 Involute gears, twenty de- gree, 201 Involute rack teeth, modified form of, 197 Involute, to draw an, 27 Iron wire, strength of, 158 Isometric projection, 48 Harmonic motion cam curve, 181 Helix, to draw a, 47 Heptagon, area of, 94 Hexagon, area of, 94 Hexagon, definition of, 14 Hexagon, to draw a, 19 Hoisting pulleys, 132 Hooke's coupling or universal joint, 263 Horse-power, 149 Horse-power of belting, 277 Horse-power of friction cone clutch, 270 Horse-power of friction disk clutch, 267 Horse-power of shafting, 274 Horse-power of steam en- gines, 81 Hunting tooth, 209 Hypocycloid, definition of, 15 Hypotenuse, definition of, 98 I Idler gear, compound, 300 Idler gear, influence of the, 299 Inclined .plane, 136 K Kirkaldy's tests on strength of materials, 157 Knuckle gears, 190 Lathe back gearing, 309 Lathe change gears, 302 Lathe change gears, simpli- fied rules for calculating, 306 Levers, 125 Levers, compound, 128 Lewis' formula for strength of gear teeth, 218 Line, definition of, 10 Line, to bisect a, 17 Lines on drawings, classes of, 55 Lock or check nuts, 248 M Machine parts, shape of, 154 Machine screws, 257 Machine steel, strength of, 158 INDEX 331 Mechanics, elements of, 120 Materials, indicating, 72 Mechanism, trains of, 297 Metric screw thread, form of, 256 Minute, definition of, 97 Moment, twisting or torsion- al, 272 Motion, Newton's laws of, 139 N Newton's laws of motion, 139 Nuts, check or lock, 248 Nuts, table of United States standard, 246 Oblique-angled triangles, 114 Octagon, area of, 94 Octagon, definition of, 14 Octagon, to draw an, 20 Oldham's coupling, 263 Oscillation, center of, 141 Paper, drawing, 8 Parallel forces, 123 Parabola, definition of, 15 Parabola, to draw a, 28 Parallelogram, area of, 92 Parallelogram, definition of, 14 Parallelogram of forces, 121 Parallel lines, 10 Parenthesis in formulas, 85 Pencils, 4 Pendulum, 141 Pens, the use of drawing, 7 Pentagon, area of, 93 Pentagon, definition of, 14 Pentagon, to draw a, 26 Perpendicular lines, 10 Perpendicular lines, to draw, 18 Pitch, circular, 205 Pitch diameters, table of, 206 Pitch, diametral, 207 Plane, definition of, 10 Plane, inclined, 136 Point, definition of, 10 Polygons, definition of, 14 Positive return cams, 173 Power transmission, screws for, 252 Presses, fly-wheels for, 289 Prism, projections of a, 34 Prism, volume of, 94 Projection, 32 Projection, isometric, 48 Pulley diameters, 281 Pulley diameters, to calcu- late, 297 Pulleys, cone, 239 Pulleys, differential, 134 Pulleys, guide, 285 Pulleys, hoisting, 132 Punches, fly-wheels for, 289 Pyramid, surface develop- ment of a, 41 Pyramid, volume of, 94 Q Quarter-turn belting, 283 Quick return device, Whit- worth, 319 Quick return motions, 313 R Rack teeth, modified form of involute, 197 Rack with epicycloidal teeth, 194 Reciprocating motion cams, 171 Resolution of forces, 123 Resultant of forces, 120 Return device, Whitworth quick, 319 Return motion, elliptic gear quick, 321 332 INDEX Return motions, quick, 313 Reversal of direction of mo- tion, to secure, 298 Right-angled triangles, 97 Safety, factor of, 151 Scales, 2 Screw cutting, gearing for, 302 Screw, differential, 138 Screw, in mechanics, 138 Screw thread, Acme stand- ard, 253 Screw thread, form of met- ric, 256 Screw thread, sharp V, 254 Screw thread, Whitworth, 255 Screw threads, drawing, 74 Screw threads, table of United States standard, 246 Screw threads, United States standard, 245 Screw threads, wrench action on, 249 Screws, bolts and studs, 243 Screws, dimensioning, 62 Screws, efficiency of, 253 Screws for power transmis- sion, 252 Screws, machine, 257 Screws, set, 256 Screws, square threaded, 251 Secant of an angle, 102 Second, definition of, 97 Sections on drawings, 66 Set-screws, 256 Shade lines, 77 Shafting, horse-power of, 274 Shafts, 272 Shafts at right angles, belt- ing between, 283 Shafts, Thurston's rule for strength of, 220 Shapers, quick return mo- tion for, 316 Sharp V-thread, 254 Shearing strength of mate- rials, 240 Shearing strength of shaft- ing, torsional, 273 Shears, fly-wheels for power, 289 Shrouded gears, 201 Sine of an angle, 101 Sines, table of, 104 Solid, definition of, 10 Speed of belting, 279 Speed ratio of gears, 220 Speed ratio of sprocket wheels, 189 Speed, to secure increase of, 297 Sphere, area and volume of, 94 Spherical sector, volume of, 94 Spherical segment, volume of, 95 Spiral, to draw a, 26 Sprocket wheels, 185 Sprocket wheels, graphical method of laying out, 187 Sprocket wheels, speed ratio of, 189 Spur gearing, calculating, 222 Spur gears, method of draw- mg, 68 Square root, 82 Square threaded screws, 251 Steel castings, strength of, 157 Steel, strength of machine, 158 Steel, strength of structural, 162 Steel wire, strength of, 158 Stepped cone pulleys, 239 Strength of gear teeth, 213 Strength of gear teeth, Lewis' formula for, 218 Strength of materials, 151 Strength of: materials, Kirk- aldy's tests on, 157 INDEX 333 Strength of materials, shear- ing, 260 Strength of shafting, tor- sional shearing, 273 Strength of shafts, twisting, 272 Stresses in castings, 162 Studs, screws and bolts, 243 Supplement angle, definition of, 11 Surface, definition of, 10 Tangent, definition of, 13 Tangent of an angle, 101 Tangent to a circle, to draw a, 19 Tangents, table of, 106 Tensile strength of materials, 158 Tension in belts, 276 Tension, machine parts sub- jected to, 154 Thickness of belts, 277 Thread, Acme standard screw, 253 Thread cutting, gearing for, 302 Thread, form of metric screw, 256 Thread, sharp V, 254 Thread, Whitworth screw, 255 Thread, drawing screw, 74 Threads, screws with square, 251 Threads, United States Standard screw, 245 Thurston's rule for strength of shafts, 220 Toothed clutches, 265 Torsional strength of shafts, 272 Trains of mechanism, 297 Transposition of formulas, 88 Triangle, area of, 91 Triangles, solution of, 96 Trigonometry, elements of, 96 Tumbler gear, 300 Twisting strength of shafts, 272 U Uniform motion cams, 165 Uniformly accelerated mo- tion cams, 176 United States standard screw thread, 245 Universal joint, 263 V-Thread, sharp, 254 Vertex of angle, definition of, 10 Views on working drawings, number of, 50 Volume of solids, 94 w Weight of fly-wheel, 291 Whitworth quick return de- vice, 319 Whitworth screw thread, 255 Width of belts, 277 Wire, strength of, 158 Work and energy, 146 Working drawings, 50 Worm gearing, 204 Worm gearing, calculating, 234 Wrench action on screw threads, 249 Wrought iron, strength of, 157 CATALOGUE OF STANDARD PRACTICAL AND SCIENTIFIC BOOKS PUBLISHED AND FOR SALE BY The Norman W, Henley Publishing Go, 132 Nassau St., New York, U. S. A. INDEX OF SUBJECTS Brazing and Soldering 3 Cams ii Charts 3 Chemistry 4 Civil Engineering 4 C oke 4 Compressed Air 4 Concrete 5 Dictionaries 5 Dies Metal Work 6 Drawing Sketching Paper 6 Electricity 7 Enameling 9 Factory Management, etc 9 Fuel 10 Gas Engines and Gas 10 Gearing and Cams u Hydraulics n Ice and Refrigeration .'....- 1 1 Inventions Patents 12 Lathe Practice 12 Liquid Air 12 Locomotive Engineering 12 Machine Shop Practice 14 Manual Training 17 Marine Engineering 17 Metal Work-Dies 6 Mining 17 Miscellaneous 18 Patents and Inventions 12 Pattern Making 18 Perfumery , 18 Plumbing , 19 Receipt Book 24 Refrigeration and Ice n Rubber 19 Saws , '. 20 Screw Cutting 20 Sheet Metal Work 20 Soldering 3 Steam Engineering 20 Steam Heating and Ventilation 22 Steam Pipes ' 22 Steel 22 Watch Making 23 Wireless Telephones 23 Any of these books will be sent prepaid to any part of the world, on receipt of price. REMIT by Draft, Postal Money Order, Express Money Order or by Registered Mail. GOOD, USEFUL BOOKS BRAZING AND SOLDERING BRAZING AND SOLDERING. By JAMES F. HOB ART. The only book that shows you just how to handle any job of brazing or soldering that comes along; tells you what mixture to use, how to make a furnace if you need one. Full of kiaks. 4th edition. 25 cents CHARTS BATTLESHIP CHART. An engraving which shows the details of a battleship as if the sides were of glass and you could see all the interior. The finest piece of work that has ever been done. So accurate that it is used at Annapolis for instruction purposes. Shows all details and gives correct name of every part. 28 x 42 inches plate paper. 50 cents BOX CAR CHART. A chart showing the anatomy of a box car, having every part of the car numbered and its proper name given in a reference list. 20 cents GONDOLA CAR CHART. A chart showing the anatomy of a gondola car, having every part of the car numbered and its proper reference name given in a reference list. 20 cents P ASSEN GER CAR CHART. A chart showing the anatomy of a passenger car, having every part of the car numbered and its proper name given in a reference list. 20 cents TRACTIVE POWER CHART. A chart v/hereby you can find the tractive power or drawbar pull of any locomotive, without making a figure. Shows what cylinders are equal, how driving wheels and steam pressure affect the power. What sized engine you need to exert a given drawbar pull or anything you desire in this line. 50 cents WESTINGHOUSE AIR-BRAKE CHARTS. Chart I. Shows (in colors) the most modern Westinghouse High Speed and Signal Equipment used on Passenger Engines, Passenger Engine Tenders, and Passenger Cars. Chart II. Shows (in colors) the Standard Westinghouse Equipment for Freight and Switch Engines, Freight and Switch Engine Tenders, and Freight Cars. Price for the set, 50 cents CHEMISTRY HENLEY'S TWENTIETH CENTURY BOOK OF RECEIPTS, FORMULAS AND PROCESSES. Edited by GARDNER D. Hiscox. The most valuable Techno-chemical Receipt Book published, including over 10,000 selected scientific chemical, technological, and practical receipts and processes. See page 24 for full description of this book. $3.00 CIVIL ENGINEERING HENLEY'S ENCYCLOPEDIA OF PRACTICAL EN- GINEERING AND ALLIED TRADES. Edited by JOSEPH G. HORNER, A.M.I., M.E. This set of five volumes contains about 2,500 pages with thousands of illustrations, including dia- grammatic and sectional drawings with full explanatory details. It covers the entire practice of Civil and Mechanical Engineering. It tells you all you want to know about engineering and tells it so simply, so clearly, so concisely that one cannot help but understand. $6.00 per volume or $25.00 for complete set of five volumes. COKE COKE MODERN COKING PRACTICE; INCLUDING THE ANALYSIS OF MATERIALS AND PRODUCTS. By T. H. BYROM, Fellow of the Institute of Chemistry, Fellow of The Chemical Society, etc., and J. E. CHRISTOPHER, Member of the Society of Chemical Industry, etc. A handbook for those engaged in Coke manufacture and the recovery of By- products. _ Fully illustrated with folding plates. The subject of Coke Manufacture is of rapidly increasing in- terest and significance, embracing as it does the recovery of valuable by-products in which scientific control is of the first importance. It has been the aim of the authors, in preparing this book, to produce one which shall be of use and benefit to those who are associated with, or interested in, the modern de- velopments of the industry. Contents: Chap. I. Introductory. Chap. II. General Classi- fication of Fuels. Chap. III. Coal Washing. Chap. IV. The Sampling and Valuation of Coal, Coke, etc. Chap. V. The Calorific Power of Coal and Coke. Chap. VI. Coke Ovens. Chap. VII. Coke Ovens, continued. Chap. VIII. Coke Ovens, continued. Chap. IX. Charging and Discharging of Coke Ovens. Chap. X. Cooling and Condensing Plant. Ch'ap. XI. Gas Ex- hausters. Chap. XII. Composition and Analysis of Ammoniacal Liquor. Chap. XIII. Working up of Ammoniacal Liquor. Chap. XIV. Treatment of Waste Gases from Sulphate Plants. Chap. XV. Valuation of Ammonium Sulphate. Chap. XVI. Direct Recovery of Ammonia from Coke Oven Gases. Chap. XVII. Surplus Gas from Coke Oven. Useful Tables. Very fully illustrated. 83.50 net COMPRESSED AIR COMPRESSED AIR IN ALL ITS APPLICATIONS By GARDNER D. Hiscox. This is the most complete book on the subject of Air that has ever been issued, and its thirty-five chapters include about every phase of the subject one can think of. It may be called an encyclopedia of compressed air. It is written by an expert, who, in its 665 pages, has dealt with the subject in a comprehensive manner, no phase of it being omitted. Over 500 illustrations, sth Edition, revised and enlarged. Cloth bound; $5.00, Half morocco, *6.5O CONCRETE ORNAMENTAL CONCRETE WITHOUT MOLDS, By A. A. HOUGHTON. The process for making ornamental concrete with- out molds, has long been held as a secret and now, for the first time, this process is given to the public. The book reveals the secret and is the only book published which explains a simple, practical method whereby the concrete worker is enabled, by employing wood and metal templates of different designs, to mold or model in concrete any Cornice, Archivolt, Column, Pedestal, Base Cap, Urn or Pier in a monolithic form right upon the job. These may be molded in units or blocks, and then built up to suit the specifications demanded. This work is fully illustrated, with detailed engravings. 83.00 POPULAR HAND BOOK FOR CEMENT AND CON- CRETE USERS, By MYRON H. LEWIS, C.E. This is a con- cise treatise of the principles and methods employed in the manufacture and use of cement in all classes of modern works. The author has brought together in this work, all the salient matter of interest to the user of concrete and its many diversified products. The matter is presented in logical and systematic order, clearly written, fully illustrated and free from involved mathematics. Everything of value to the concrete user is given. Among the chapters contained in the book are: I. Historical Development of the Uses of Cement and Concrete. II. Glossary of Terms employed in Cement and Concrete work. III. Kinds of Cement employed in Construction. IV. Limes, Ordinary and Hydraulic. V. Lime Plasters. VI. Natural Cements. VII. Portland Cements. VIII. Inspection and Testing. IX. Adul- teration; or Foreign Substances in Cement. X. Sand, Gravel and Broken Stone. XI. Mortar. XII. Grout. XIII. Con- crete (Plain). XIV. Concrete (Reinforced). XV. Methods and Kinds of Reinforcements. XVI. Forms for Plain and Re- inforced Concrete. XVII. Concrete Blocks. XVIII. Arti- ficial Stone. XIX. Concrete Tiles. XX. Concrete Pipes and Conduits. XXI. Concrete Piles. XXII. Concrete Buildings. XXIII. Concrete in Water Works. XXIV. Concrete in Sewer Works. XXV. Concrete in Highway Construction. XXVI. Concrete Retaining Walls. XXVII. Concrete Arches and Abutments. XXVIII. Concrete in Subway and Tunnels. XXIX. Concrete in Bridge Work. XXX. Concrete in Docks and Wharves. XXXI. Concrete Construction under Water. XXXII. Concrete on the Farm. XXXIII. Concrete Chimneys. XXXIV. Concrete for Ornamentation. XXXV. Concrete Mausoleums and Miscellaneous Uses. XXXVI. Inspection for Concrete Work. XXX VII. Waterproofing Concrete Work. XXXVIII. Coloring and Painting Concrete Work. XXXIX. Method of Finishing Concrete Surfaces. XL. Specifications and Estimates for Concrete Work. $3.50 DICTIONARIES STANDARD ELECTRICAL DICTIONARY. By T. O'CoNOR SLOANE. An indispensable work to all interested in electrical science. Suitable alike for the student and profession- al. A practical hand-book of reference containing definitions of about 5,000 distinct words, terms and phrases. The defini- tions are terse and concise and include every term used in electri- cal science. Recently issued. An entirely new edition. Should be in the possession of all who desire to keep abreast with the progress of this branch of science. Complete, concise and con- venient. 682 pages 393 illustrations. 83.00 DIES METAL WORK DIES, THEIR CONSTRUCTION AND USE FOR THE MODERN WORKING OF SHEET METALS. By J. V. WOODWORTH. A new book by a practical man, for those who wish to know the latest practice in the working of sheet metals. It shows how dies are designed, made and used, and those who are engaged in this line of work can secure many valuable sug- gestions. $3.00 PUNCHES, DIES AND TOOLS FOR MANUFACTUR- ING IN PRESSES. By J. V. WOODWORTH. An encyclo- pedia of die-making, punch-making, die-sinking, sheet-metal working, and making of special tools, subpresses, devices and mechanical combinations for punching, cutting, bending, form- ing, piercing, drawing, compressing, and assembling sheet- metal parts and also articles of other materials in machine tools. This is a distinct work from the author's book entitled "Dies; Their Construction and Use." 500 pages, 700 engrav- ings. $4.00 DRAWING SKETCHING PAPER LINEAR PERSPECTIVE SELF-TAUGHT. By HERMAN T. C. KRAUS. This work gives the theory and practice of linear perspective, as used in architectural, engineering, and mechanical drawings. Persons taking up the study of the subject by them- selves, without the aid of a teacher, will be able by the use of the instruction given to readily grasp the subject, and by reason- able practice become good perspective draftsmen. The arrange- ment of the book is good; the plate is on the left-hand, while the descriptive text follows on the opposite page, so as to be readily referred to. The drawings are on sufficiently large scale to show the work clearly and are plainly figured. The whole work makes a very complete course on perspective drawing, and will be found of great value to architects, civil and mechanical engineers, patent attorneys, art designers, engravers, and draftsmen. $2.50 PRACTICAL PERSPECTIVE. By RICHARDS and COLVIN. Shows just how to make all kinds of mechanical drawings in the only practical perspective isometric. Makes everything plain so that any mechanic can understand a sketch or drawing in this way. Saves time in the drawing room and mistakes in the shops. Contains practical examples of various classes of work. 50 cents SELF-TAUGHT MECHANICAL DRAWING AND ELE- MENTARY MACHINE DESIGN. By F. L. SYLVESTER, M.E., Draftsman, with additions by Erik Oberg, associate editor of "Machinery." A practical elementary treatise on Mechanical Drawing and Machine Design, comprising the first principles of geometric and mechanical drawing, workshop _ mathematics, mechanics, strength of materials and the calculation and design of machine details, compiled for the use of practical mechanics and young draftsmen. $2.00 A NEW SKETCHING PAPER. A new specially ruled paper to enable you to make sketches or drawings in isometric per- spective without any figuring or fussing. It is being used for shop details as well as for assembly drawings, as it makes one sketch do the work of three, and no workman can help seeing just what is wanted. Pads of 40 sheets, 6x9 inches, 25 cents. Pads of 40 sheets, 9x12 inches, 50 cents ELECTRICITY ARITHMETIC OF ELECTRICITY. By Prof. T. O'CoNOR SLOANE. A practical treatise on electrical calculations of all kinds reduced to a series of rules, all of the simplest forms, and involving only ordinary arithmetic; each rule illustrated by one or more practical problems, with detailed solution of each one. This book is classed among the most useful works pub- lished on the science of electricity Covering as it does the mathe- matics of electricity in a manner that will attract the attention of those who are not familiar with algebraical formulas. 160 pages. $1.00 COMMUTATOR CONSTRUCTION. By WM. BAXTER, JR. The business end of any dynamo or motor of the direct current type is the commutator. This book goes into the de- signing, building, and maintenance of commutators, shows how to locate troubles and how to remedy them; everyone who fusses with dynamos needs this. 25 cents DYNAMO BUILDING FOR AMATEURS, OR HOW TO CONSTRUCT A FIFTY WATT DYNAMO. By ARTHUR J. WEED, Member of N. Y. Electrical Society. This book is a practical treatise showing in detail the construction of a small dynamo or motor, the entire machine work of which can be done on a small foot lathe. Dimensioned working drawings are given for each piece of machine work and each operation is clearly described. This machine when used as a dynamo has an output of fifty watts; when used as a motor it will drive a small drill press or lathe. It can be used to drive a sewing machine on any and all ordinary work. The book is illustrated with more than sixty original engrav- ings showing the actual construction of the different parts. Paper. Paper 50 cents Cloth 81.00 ELECTRIC FURNACES AND THEIR INDUSTRIAL APPLICATIONS. By J.WRIGHT. This is a book which will prove of interest to many classes of people; the manufacturer who desires to know what product .can be manufactured success- fully in the electric furnace, the chemist who wishes to post himself on the electro-chemistry, and the student of science who merely looks into the subject from curiosity. 288 pages. $3.00 ELECTRIC LIGHTING AND HEATING POCKET BOOK. By SYDNEY F. WALKER. This book puts in conven- ient form useful information regarding the apparatus which is likely to be attached to the mains of an electrical company. Tables of units and equivalents are included and useful electrical laws and formulas are stated. 43 8 pages, 3 oo engravings. $3.00 ELECTRIC TOY MAKING, DYNAMO BUILDING, AND ELECTRIC MOTOR CONSTRUCTION. This work treats of the making at home of electrical toys, electrical apparatus, motors, dynamos, and instruments in general, and is designed to bring within the reach of young and old the manufacture of gen- uine and useful electrical appliances. 185 pages. Fully illus- trated. $1.00 ELECTRIC WIRING, DIAGRAMS AND SWITCH- BOARDS. By NEWTON HARRISON. This is the only complete work issued snowing and telling you what you should know about direct and alternating current wiring. It is a ready reference. The work is free from advanced technicalities and mathematics. Arithmetic being used throughout. It is in every respect a handy, well-written, instructive, comprehensive volume on wiring for the wireman, foreman, contractor or elec- trician. 272 pages, 105 illustrations. 81.50 'ELECTRICIAN'S HANDY BOOK. By PROF. T. O'CpNOR SLOANE. This work is intended for the practical electrician, who has to make things go. The entire field of Electricity is covered within its pages. It contains no useless theory; every- thing is to the point. It teaches you just what you should know about electricity. It is the standard work published on the subject. Forty-one chapters, 610 engravings, handsomely bound in red leather with titles and edges in gold. $3.50 ELECTRICITY IN FACTORIES AND WORKSHOPS, ITS COST AND CONVENIENCE. By ARTHUR P. HASLAM. A practical book for power producers and power users showing what a convenience the electric motor, in its various forms, has become to the modern manufacturer. It also deals with the conditions which determine the cost of electric driving, and compares this with other methods of producing and utilizing power. 312 pages. Very fully illustrated. $2.50 ELECTRICITY SIMPLIFIED. By PROF. T. O'CoNOR SLOANE. The object of "Electricity Simplified" is to make the subject as plain as possible and to show what the modern con- ception of electricity is; to show how two plates of different metals immersed in acid can send a message around the globe; to explain how a bundle of copper wire rotated by a steam engine can be the agent in lighting our streets, to tell what the volt, ohm and ampere are, and what high and low tension mean; and to answer the questions that perpetually arise in the mind in this age of electricity. 172 pages. Illustrated. $1.00 HOW TO BECOME A SUCCESSFUL ELECTRICIAN. By PROF. T. O'CoNOR SLOANE. An interesting book from cover to cover. Telling in simplest language the surest and easiest way to become a successful electrician. The studies to be followed, methods of work, field of operation and the requirements of the successful electrician are pointed out and fully explained. 202 pages. Illustrated. $1.00 MANAGEMENT OF DYNAMOS. By LUMMIS-PATER- SON. A handbook of theory and practice. This work is arranged in three parts. The first part covers the elementary theory of the dynamo. The second part, the construction and action of the different classes of dynamos in common use are described; while the third part relates to such matters as affect the prac- tical management and working of dynamos and motors. 292 pages, 117 illustrations. $1.50 STANDARD ELECTRICAL DICTIONARY. By Prof. T. O'CoNOR SLOANE. A practical handbook of reference contain- ing definitions of about 5,000 distinct words, terms and phrases. The definitions are terse and concise and include every term used in electrical science. 682 pages, 393 illustrations. $3.00 8 SWITCHBOARDS. By WILLIAM BAXTER, JR. This book appeals to every engineer and electrician who wants to know the practical side of things. All sorts and conditions of dynamos, connections and- circuits are shown by diagram and illustrate just how the switchboard should be connected. Includes direct and alternating current boards, also those for arc lighting, in- candescent, and power circuits. Special treatment on high voltage boards for power transmission. 190 pages. Illustrated. 81.50 TELEPHONE CONSTRUCTION, INSTALLATION, WIRING, OPERATION AND MAINTENANCE. By W. H. RADCLIFFE and H. C. CUSHING. This book gives the principles of construction and operation of both the Bell and Independent instruments; approved methods of installing and wiring them; the means of protecting them from lightning and abnormal cur- rents; their connection together for operation as series or bridg- ing stations; and rules for their inspection and maintenance. Line wiring and the wiring and operation of special telephone systems are also treated. 180 pages, 125 illustrations. 81.00 WIRING A HOUSE. By HERBERT PRATT. Shows a house already built; tells just how to start about wiring it. Where to begin; what wire to use; how to run it according to insurance rules, in fact just the information you need. Directions apply equally to a shop. Fourth edition. 35 cents WIRELESS TELEPHONES AND HOW THEY WORK. By JAMES ERSKINE-MURRAY. This work is free from elaborate details and aims at giving a clear survey of the way in which Wireless Telephones work. It is intended for amateur workers and for those whose knowledge of Electricity is slight. Chap- ters contained: How We Hear Historical The Conversion of Sound into Electric Waves Wireless Transmission The Pro- duction of Alternating Currents of High Frequency How the Electric Waves are Radiated and Received The Receiving Instruments Detectors Achievements and Expectations Glossary of Technical Work. Cloth. 81.00 ENAMELING HENLEY'S TWENTIETH CENTURY RECEIPT BOOK. Edited by GARDNER D. Hiscox. A work of 10,000 practical receipts, including enameling receipts for hollow ware, for metals, for signs, for china and porcelain, for wood, etc. Thor- ough and practical. See page 24 for full description of this book. 3.00 FACTORY MANAGEMENT, ETC. MODERN MACHINE SHOP CONSTRUCTION, EQUIP- MENT AND MANAGEMENT. By O. E. PERRIGO, M.E. A work designed for the practical and every-day use of the Archi- tect who designs, the Manufacturers who build, the Engineers who plan and. equip, the Superintendents who organize and direct, and for the information of every stockholder, director, officer, accountant, clerk, superintendent, foreman, and work- man of the modern machine shop and manufacturing plant of Industrial America. 85.00 FUEL COMBUSTION OF COAL AND THE PREVENTION OF SMOKE. By WM. M. BARR. To be a success a fireman must be "Light on Coal." He must keep his fire in good con- dition, and prevent, as far as possible, the smoke nuisance. To do this, he should know how coal burns, how smoke is formed and the proper burning of fuel to obtain the best results. He can learn this, and more too, from Barr's "Combustion of Coal." It is an absolute authority on all questions relating to the Firing of a Locomotive. Nearly 350 pages, fully illustrated. 81.00 SMOKE PREVENTION AND FUEL ECONOMY. By BOOTH and KERSHAW. As the title indicates, this book of 197 pages and 75 illustrations deals with the problem of complete combustion; which it treats from the chemical and mechanical standpoints, besides pointing out the economical and humani- tarian aspects of the question. S2.5O GAS ENGINES AND GAS CHEMISTRY OF GAS MANUFACTURE. By H. M. ROYLES. A practical treatise for the use of gas engineers, gas managers and students. Including among its contents Prepa- rations of Standard Solutions, Coal, Furnaces, Testing and Regulation. Products of Carbonization. Analysis of Crude Coal Gas. Analysis of Lime. Ammonia. Analysis of Oxide of Iron. Naphthalene. Analysis of Fire-Bricks and Fire-Clay . Weldom and Spent Oxide. Photometry and Gas Testing. Carbur- etted Water Gas. Metropolis Gas. Miscellaneous Extracts. Useful Tables. $4.50 GAS ENGINE CONSTRUCTION, Or How to Build a Half- Horse-power Gas Engine. By PARSELL and WEED. A prac- tical treatise describing the theory and principles of the action of gas engines of various types, and the design and construction of a half-horse-power gas engine, with illustrations of the work in actual progress, together with dimensioned working drawings giv- ing clearly the sizes of the various details. 300 pages. $2.50 GAS, GASOLINE, AND OIL, ENGINES. By GARDNER D. Hiscox. Just issued, i8th revised and enlarged edition. Every user of a gas engine needs this book. Simple, instructive, and right up-to-date. The only complete work on the subject. Tells all about the running and management of gas, gasoline and oil engines as designed and manufactured in the United States. Explosive motors for stationary, marine and vehicle power are fully treated, together with illustrations of their parts and tabu- lated sizes, also their care and running are included. Electric Ignition by Induction Coil and Jump Sparks are fully explained and illustrated, including valuable information on the testing for economy and power and the erection of power plants. The special information on PRODUCER and SUCTION GASES in- cluded cannot fail to prove of value to all interested in the gen- eration of producer gas and its utilization in gas engines. The rules and regulations of the Board of Fire Underwriters in regard to the installation and management of Gasoline Motors is given in full, suggesting the safe installation of explosive motor power. A list of United States Patents issued on Gas, ^Gasoline and Oil Engines and their adjuncts from 1875 to date is included. 484 pages. 410 engravings. S3. 50 net MODERN GAS ENGINES AND PRODUCER GAS PLANTS. By R. E. MATHOT, M.E. A practical treatise of 320 pages, fully illustrated by 175 detailed illustrations, setting forth the principles of gas engines and producer design, the selec- tion and installation of an engine, conditions of perfect opera- tion, producer-gas engines and their possibilities, the care of gas engines and producer-gas plants, with a chapter on volatile hydrocarbon and oil engines. This book has been endorsed by Dugal Clerk as a most useful work for all interested in Gas Engine installation and Prodxicer Gas. 82.50 GEARING AND CAMS BEVEL GEAR TABLES. By D. AG. ENGSTROM. No one who has to do with bevel gears in any way should be without this book. The designer and draftsman will find it a great con- venience, while to the machinist who turns up the blanks or cuts the teeth, it is invaluable, as all needed dimensions are given and no fancy figuring need be done. 81. OO CHANGE GEAR DEVICES. By OSCAR E. PERRIGO. A book for every designer, draftsman and mechanic who is inter- ested in feed changes for any kind of machines. This shows what has been done and how. Gives plans, patents and all information that you need. Saves hunting through patent records and rein- venting old ideas. A standard work of reference. 81.00 DRAFTING OF CAMS. By Louis ROUILLION. The laying out of cams is a serious problem unless you know how to go at it right. This puts you on the right road for practically any kind of cam you are likely to run up against. 25 cents HYDRAULICS HYDRAULIC ENGINEERING. By GARDNER D. Hiscox. A treatise on the properties, power, and resources of water for all purposes. Including the measurement of streams; the flow of water in pipes or conduits; the horse-power of falling water; turbine and impact water-wheels; wave-motors, centrifugal, reciprocating, and air-lift pumps. With 300 figures and dia- grams and 36 practical tables. 320 pages. 84.00 ICE AND REFRIGERATION POCKET BOOK OF REFRIGERATION AND ICE MAK- ING, By A. J. WALLIS-TAYLOR. This is one of the latest and most comprehensive reference books published on the subject of refrigeration and cold storage. It explains the properties and refrigerating effect of the different fluids in use, the manage- ment of refrigerating machinery and the construction and insula- tion of cold rooms with their required pipe surface for different degrees of cold; freezing mixtures and non-freezing brines, temperatures of cold rooms for all kinds of provisions, cold storage charges for all classes of goods, ice making and storage of ice, data and memoranda for constant reference by refrigerating engineers, with nearly one hundred tables containing valuable references to every fact and condition required in the installment and operation of a refrigerating plant. 81.50 II INVENTIONS PATENTS INVENTOR'S MANUAL, HOW TO MAKE A PATENT PAY. This is a book designed as a guide to inventors in per- fecting their inventions, taking out their patents, and disposing of them. It is not in any sense a Patent Solicitor's Circular, nor a Patent Broker's Advertisement. No advertisements of any description appear in the work. It is a book containing a quarter of a century's experience of a successful inventor, together with notes based upon the experience of many other inventors. $J .00 LATHE PRACTICE MODERN AMERICAN LATHE PRACTICE. By OSCAR E. PERRIGO. An up-to-date book on American Lathe Work, describing and illustrating the very latest practice in lathe and boring-mill operations, as well as the construction of and latest devel9pments in the manufacture of these important classes of machine tools. 300 pages, fully illustrated. 83.50 PRACTICAL METAL TURNING. By JOSEPH G. HORNER. A work of 404 pages, fully illustrated, covering in a comprehen- sive manner the modern practice of machining metal parts in the lathe, including the regular engine lathe, its essential design, its uses, its tools, its attachments, and the manner of holding the work and performing the operations. The modernized engine lathe, its methods, tools, and great range of accurate work. The Turret Lathe, its tools, accessories and methods of performing its functions. Chapters on special work, grinding, tool holders, speeds, feeds, modern tool steels, etc., etc. $3.50 TURNING AND BORING TAPERS. By FRED H. COL- VIN. There are two ways to turn tapers; the right way and one other. This treatise has to do with the right way; it tells you how to start the work properly, how to set the lathe, what tools to use and how to use them, and forty^and one other little things that you should know. Fourth edition. 25 cents LIQUID AIR LIQUID AIR AND THE LIQUEFACTION OF GASES. By T. O'CoNOR SLOANE. Theory, history, biography, practical applications, manufacture. 365 pages. Illustrated. $2.00 LOCOMOTIVE ENGINEERING AIR-BRAKE CATECHISM. By ROBERT H. BLACKALL. This book is a standard text book. It covers the Westinghouse Air-Brake Equipment, including the No. 5 and the No. 6 E T Locomotive Brake Equipment; the K (Quick-Service) Triple Valve for Freight Service; and the Cross-Compound Pump. The operation of all parts of the apparatus is explained in detail, and a practical way of finding their peculiarities and defects, with a proper remedy, is given. It contains 2,000 questions with their answers, which will enable any railroad man to pass any examination on the subject of Air Brakes. Endorsed and used by air-brake instructors and examiners on nearly every rail- road in the United States. 2 3 d Edition. 380 pages, fully illustrated with folding plates and diagrams. $2.00 AMERICAN COMPOUND LOCOMOTIVES. By FRED H. COLVIN. The most complete book on compounds published. Shows all types, including the balanced compound. Makes everything clear by many illustrations, and shows valve setting, breakdowns and repairs. 142 pages. $1.00 APPLICATION OF HIGHLY SUPERHEATED STEAM TO LOCOMOTIVES. By ROBERT GARBE. A practical book. Contains special chapters on Generation of Highly Superheated Steam; Superheated Steam and the Two-Cylinder Simple Engine; Compounding and Superheating; Designs of Locomotive Superheaters; Constructive Details of Locomotives using Highly Superheated Steam; Experimental and Working Results. Illus- trated with folding plates and tables. 82.50 COMBUSTION OF COAL AND THE PREVENTION OF SMOKE. By WM. M. BARR. To be a success a fireman must be "Light on Coal." He must keep his fire in good con- dition, and prevent as far as possible, the smoke nuisance. To do this, he should know how coal burns, how smoke is formed and the proper burning of fuel to obtain the best results. He can learn this, and more too, from Barr's "Combination of Coal." It is an absolute authority on all questions relating to the Firing of a Locomotive. Nearly 350 pages, fully illustrated. $1.00 LINK MOTIONS, VALVES AND VALVE SETTING. By FRED H. COLVIN, Associate Editor of "American Machinist. A handy book that clears up the mysteries of valve setting. Shows the different valve gears in use, how they work, and why. Piston and slide valves of different types are illustrated and explained. A book that every railroad man in the motive- power department ought to have. Fully illustrated. 60 cents. LOCOMOTIVE BOILER CONSTRUCTION. By FRANK A. KLEINHANS. The only book showing how locomotive boilers are built in modern shops. Shows all types of boilers used; gives details of construction; practical facts, such as life of riveting punches and dies, work done per day, allowance for bending and flanging sheets and other data that means dol- lars to any railroad man. 421 pages, 334 illustrations. Six folding plates. $3.00 LOCOMOTIVE BREAKDOWNS AND THEIR REM- EDIES. By GEO. L. FOWLER. Revised by Wm. W. Wood, Air-Brake Instructor. Just issued 1910 Revised pocket edition. It is put of the question to try and tell you about every subject that is covered in this pocket edition of Locomotive Breakdowns. Just imagine all the common troubles that an engineer may ex- pect to happen some time, and then add all of the unexpected ones, troubles that could occur, but that you had never thought about, and you will find that they are all treated with the very best methods of repair. Walschaert Locomotive Valve Gear Troubles, Electric Headlight Troubles, as well as Questions and Answers on the Air Brake are all included. 294 pages. Fully illustrated. $1.00 LOCOMOTIVE CATECHISM. By ROBERT GRIMSHAW. 27th revised and enlarged edition. This may well be called an encyclopedia of the locomotive. Contains over 4,000 examina- tion questions with their answers, including among them those asked at the First, Second and Third year's Examinations. 825 pages, 437 illustrations and 3 folding plates. $2.50 13 NEW YORK AIR-BRAKE CATECHISM. By ROBERT H. BLACKALL. This is a complete treatise on the New York Air-Brake and Air-Signalling Apparatus, giving a detailed de- scription of all the parts, their operation, troubles, and the methods of locating and remedying the same. 200 pages, fully illustrated. 81.00 POCKET-RAILROAD DICTIONARY AND VADE ME- CU!\I. ^ By FRED H. COLVIN, Associate Editor "American Machinist." Different from any book you ever saw. Gives clear and concise information on just the points you are interested in. It's really a pocket dictionary, fully illustrated, and so arranged that you can find just what you want in a second without an index. Whether you are interested in Axles or Acetylene; Com- pounds or Counter Balancing; Rails or Reducing Valves; Tires or Turntables, you'll find them in this little book. It's very complete. Flexible cloth cover, 200 pages. 81.00 TRAIN RULES AND DESPATCHING. By H. A. DALBY. Contains the standard code for both single and double track and explains how trains are handled under all conditions. Gives all signals in colors, is illustrated wherever necessary, and the most complete book in print on this important subject. Bound in fine seal flexible leather. 221 pages. 81.50 WALSCHAERT LOCOMOTIVE VALVE GEAR. By WM. W. WOOD. If you would thoroughly understand the Walschaert Valve Gear, you should possess a copy of this book. The author divides the subject into four divisions, as follows: I. Analysis of the gear. II. Designing and erecting of the gear III. Advantages of the gear. IV. Questions and answers re lating to the Walschaert Valve Gear. This book is specially valu- able to those preparing for promotion. Nearly 200 pages. $1.50 WESTINGHOUSE E T AIR-BRAKE INSTRUCTION POCKET BOOK CATECHISM. By WM. W. WOOD, Air-Brake Instructor. A practical work containing examination questions and answers on the E T Equipment. Covering what the E T Brake is. How it should be operated. What to do when de- fective. Not a question can be asked of the engineman up for promotion on either the No. 5 or the No. 6 E T equipment that is not asked and answered in the book. If you want to thor- oughly understand the E T equipment get a copy of this book. It covers every detail. Makes Air-Brake troubles. and examina- tions easy. Fully illustrated with colored plates, showing various pressures. 82.00 MACHINE SHOP PRACTICE AMERICAN TOOL MAKING AND INTERCHANGE- ABLE MANUFACTURING. ^ By J. V. WOODWORTH. A practical treatise on the designing, constructing, use, and in- stallation of tools, jigs, fixtures, devices, special appliances, sheet-metal working processes, automatic mechanisms, and labor-saving contrivances; together with their use in the lathe milling machine, turret lathe, screw machine, boring mill, power press, drill, sub press, drop hammer, etc., for the working of metals, the production of interchangeable machine parts, and the manufacture of repetition articles of metal. 560 pages, 600 illustrations. *4.0O HENLEY'S ENCYCLOPEDIA OF PRACTICAL EN- GINEERING AND ALLIED TRADES. Edited by JOSEPH G. HORNER. A.M.I.Mech.I. This work covers the entire prac- tice of Civil and Mechanical Engineering. The best known ex- perts in all branches of engineering have contributed to these volumes. The Cyclopedia is admirably well adapted to the needs of the beginner and the self-taught practical man, as well as the mechanical engineer, designer, draftsman, shop superintendent, foreman and machinist. It is a modern treatise in five volumes. Handsomely bound in Half Morocco, each volume containing nearly 500 pages, with thousands of illustrations, including diagrammatic and sectional drawings with full explanatory details. $35.00 for the com- plete set of five volumes. $6.00 per volume, when ordered singly. MACHINE SHOP ARITHMETIC. By COLVIN-CHENEY. Most popular book for shop men. Shows how all shop problems are worked out and "why." Includes change gears for cutting any threads; drills, taps, shink and force fits; metric system of measurements and threads. Used by all classes of mechanics and for instruction of Y. M. C. A. and other schools. Fifth edition. 131 pages. 50 cents MECHANICAL MOVEMENTS, POWERS, AND DE- VICES. By GARDNER D. Hiscox. This is a collection of 1890 engravings of different mechanical motions and appliances, ac- companied by appropriate text, making it a book of great value to the inventor, the draftsman, and to all readers with mechanical tastes. The book is divided into eighteen sections or chapters in which the subject matter is classified under the following heads: Mechanical Powers, Transmission of Power, Measurement of Power, Steam Power, Air Power Appliances, Electric Power and Construction, Navigation and Roads, Gearing, Motion and Devices, Controlling Motion, Horological, Mining, Mill and Factory Appliances, Construction and Devices, Drafting Devices, Miscellaneous Devices, etc. nth edition. 400 octavo pages. $3.50 MECHANICAL APPLIANCES, MECHANICAL MOVE- MENTS AND NOVELTIES OF CONSTRUCTION. By GARDNER D. Hiscox. This is a supplementary volume to the one upon mechanical movements. Unlike the first volume, which is more elementary in character, this volume contains illustrations and descriptions of many combinations of motions and of mechanical devices and appliances found in different lines of Machinery. Each device being shown by a line drawing with a description showing its working parts and the method of opera- tion. From the multitude of devices described, and illustrated, might be mentioned, in passing, such items as conveyors and elevators, Prony brakes, thermometers, various types of boilers, solar engines, oil-fuel burners, condensers, evaporators, Corliss and other valve gears, governors, gas engines, water motors of various descriptions, air ships, motors and dynamos, automobile and motor bicycles, railway block signals, car couples, link and gear motions, ball bearings, breech block mechanism for heavy guns, and a large accumulation of others of equal importance. 1,000 specially made engravings. 396 octavo pages. $2.50 These two volumes sell for $2.50 each, but when the twQ volumes are order ed at one time from us, we send them prepaid to any address in the world, on receipt of $4.00. You save $i by ordering the two volumes of Mechanical Movements at one time. 15 MODERN MACHINE SHOP CONSTRUCTION, EQUIP- MENT AND MANAGEMENT. By OSCAR E. PERRIGO. The only work published that describes the Modern Machine Shop or Manufacturing Plant from the time the grass is growing on the site intended for it until the finished product is shipped. Just the book needed by those contemplating the erection of modern shop buildings, the rebuilding and reorganization of old ones, or the introduction of Modern Shop Methods, Time and Cost Systems. It is a book written and illustrated by a prac- tical shop man for practical shop men who are too busy to read theories and want facts. It is the most complete all-around book of its kind ever published. 400 large quarto pages, 225 original and specially-made illustrations. $5.00 MODERN MACHINE SHOP TOOLS; THEIR CON- STRUCTION, OPERATION, AND MANIPULATION. By W. H. VANDERVOORT. A work of 555 pages and 673 illustra- tions, describing in every detail the construction, operation, and manipulation of both Hand and Machine Tools. Includes chapters on filing, fitting, and scraping surfaces; on drills, ream- ers, taps, and dies; the lathe and its tools; planers, shapers, and their tools; milling machines and cutters; gear cutters and gear cutting; drilling machines and drill work; grinding ma- chines and their work; hardening and tempering; gearing, belting and transmission machinery; useful data and tables. $4.00 THE MODERN MACHINIST. By JOHN T. USHER. This book might be called a compendium of shop methods, showing a variety of special tools and appliances which will give new ideas to many mechanics from the superintendent down to the man at the bench. It will be found a valuable addition to any machin- ist's library and should be consulted whenever a new or difficult job is to be done, whether it is boring, milling, turning, or plan- ing, as they are all treated in a practical manner. Fifth edition. 320 pages, 250 illustrations. $2.50 MODERN MECHANISM. Edited by PARK BENJAMIN. A practical treatise on machines, motors and the transmission of power, being a complete work and a supplementary volume to Appleton's Cyclopedia of Applied Mechanics. Deals solely with the principal and most useful advances of the past few years. 959 pages containing over 1,000 illustrations; bound in half morocco. $4.00 MODERN MILLING MACHINES : THEIR DESIGN, CONSTRUCTION AND OPERATION. By JOSEPH G. HORNER. This book describes and illustrates the Milling Ma- chine and its work in such a plain, clear, and forceful manner, and illustrates the subject so clearly and completely, that the up-to-date machinist, student, or mechanical engineer can not afford to do without the valuable information which it contains. It describes not only the early machines of this class, but notes their gradual development into the splendid machines of the present day, giving the design and construction of the various types, forms, and special features produced by prominent manufacturers, American and foreign. 304 pages, 300 illustra- tions. $4.00 " SHOP KINKS." By ROBERT GRIMSHAW. This shows special methods of doing work of various kinds, and reducing cost of production. Has hints and kinks from some of the largest shops in th'is country and Europe. You are almost sure to find some that apply to your work, and in such a way as to save time and trouble. 400 pages. Fourth edition. $2.50 16 TOOLS FOR MACHINISTS AND WOOD WORKERS, INCLUDING INSTRUMENTS OF MEASUREMENT. By JOSEPH G. HORNER. A practical treatise of 340 pages, fully illustrated and comprising a general description and classifica- tion of cutting tools and tool angles, allied cutting tools for machinists and woodworkers; shearing tools; scraping tools; saws; milling cutters; drilling and boring tools; taps and dies; punches and hammers; and the hardening, tempering and grinding of these tools. Tools for measuring and testing work, including standards of measurement; surface plates; levels; surface gauges; dividers; calipers; verniers; micrometers; snap, cylindrical and limit gauges; screw thread, wire and reference gauges, indicators, templets, etc. 83. 50 MANUAL TRAINING ECONOMICS OF MANUAL, TRAINING. By Louis ROUILLION. The only book that gives just the information needed by all interested in manual training, regarding buildings, equipment and supplies. Shows exactly what is needed for all grades of the work from the Kindergarten to the High and Nor- mal School. Gives itemized lists of everything needed and tells just what it ought to cost. Also shows where to buy supplies. $1.50 MARINE ENGINEERING MARINE ENGINES AND BOILERS, THEIR DESIGN AND CONSTRUCTION. By DR. G. BAUER, LESLIE S. ROBERTSON, and S. BRYAN DONKIN. This work is clearly written, thoroughly systematic, theoretically sound; while the character of its plans, drawings, tables, and statistics is without reproach. The illustrations are careful reproductions from actual working drawings, with some well-executed photographic views of completed engines and boilers. $9.00 net MINING *ORE DEPOSITS OF SOUTH AFRICA WITH A CHAPTER ON HINTS TO PROSPECTORS. By J. P. JOHN- SON. This book gives a condensed account of the ore-deposits at present known in South Africa. It is also intended as a guide to the prospector. Only an elementary knowledge of geology and some mining experience are necessary in order to under- stand this work. With these qualifications, it will materially assist one in his search for metalliferous mineral occurrences and, so far as simple ores are concerned, should enable one to form some idea of the possibilities of any they may find. Among the chapters given are: Titaniferous and Chromif- erous Iron Oxides Nickel Copper Cobalt Tin Molyb- denum Tungsten Lead Mercury Antimony I r o n Hints to Prospectors. Illustrated. $2.00 PRACTICAL COAL MINING. By T. H. COCKIN. An im- portant work, containing 428 pages and 213 illustrations, com- plete with practical details, which will intuitively impart to the reader, not only a general knowledge of the principles of coal mining, but also considerable insight into allied subjects. The treatise is positively up to date in every instance, and should be in the hands of every colliery engineer, geologist, mine operator, superintendent, foreman, and all others who are in- terested in or connected with the industry. $2.50 17 PHYSICS AND CHEMISTRY OF MINING. By T. H. BYROM. A practical work for the use of all preparing for ex- aminations in mining or qualifying for colliery managers' cer- tificates. The aim of the author in this excellent book is to place clearly before the reader useful and authoritative data which will render him valuable assistance in his studies. The only work of its kind published. The information incorporated in it will prove of the greatest practical utility to students, mining en- gineers, colliery managers, and all others who are specially in- terested in the present-day treatment of mining problems. 160 pages. Illustrated. $3.00 MISCELLANEOUS BRONZES. Henley's Twentieth Century Receipt Book con- tarns many practical formulas on bronze casting, imitation bronze, bronze polishes, renovation of bronze. See page 24 for full description of this book. 83.00 EMINENT ENGINEERS. By DWIGHT GODDARD. Every- one who appreciates the effect of such great inventions as the Steam Engine, Steamboat, Locomotive, Sewing Machine, Steel Working, and other fundamental discoveries, is interested in knowing a little about the men who made them and their achieve- ments. Mr. Goddard has selected thirty-two of the world's engineers who have contributed most largely to the advancement of our civilization by mechanical means, giving only such facts as are of general interest and in a way which appeals to all, whether mechanics or not. 280 pages, 35 illustrations. $1.50 LAWS OF BUSINESS, By THEOPHILUS PARSONS, LL.D. The Best Book for Business Men ever Published. Treats clearly of Contracts, Sales, Notes, Bills of -Exchange, Agency, Agree- ment, Stoppage in Transitu, Consideration, Limitations, Leases, Partnership, Executors, Interest, Hotel Keepers, Fire and Life Insurance, Collections, Bonds, Frauds, Receipts, Patents, Deeds, Mortgages, Liens, Assignments, Minors, Married Women, Arbi- tration, Guardians, Wills, etc. Three Hundred Approved Forms are given. Every Business Man should have a copy of this book for ready reference. . The book is bound in full sheep, and Con- tains 864 Octavo Pages. Our special price. $3.50 PATTERN MAKING PRACTICAL PATTERN MAKING. By F. W. BARROWS. This is a very complete and entirely practical treatise on the subject of pattern making, illustrating pattern work in wood and metal. From its pages you are taught just what you should know about pattern making. It contains a detailed description of the materials used by pattern makers, also the tools, both those for hand use, and the more interesting machine tools; hav- ing complete chapters on The Band Saw, The Buzz Saw, and The Lathe. Individual patterns of many different kinds are fully illustrated and described, and the mounting of metal patterns on plates for molding machines is included. $3.00 PERFUMERY HENLEY'S TWENTIETH CENTURY BOOK OF RE- CEIPTS, FORMULAS AND PROCESSES. Edited by G. D. Hiscox. The most valuable Techno-Chemical Receipt Book published. Contains over 10,000 practical Receipts many of which will prove of special value to the perfumer, a mine of in- formation, up to date in every respect. Cloth, $3.OO; half morocco. See page 24 for full description of this book. $4.00 18 PERFUMES AND THEIR PREPARATION. By G. W. ASKINSON, Perfumer. A comprehensive treatise, in which there has been nothing omitted that could be of value to the Perfumer. Complete directions for making handkerchief per- fumes, smelling-salts, sachets, fumigating pastilles; preparations for the care of the skin, the mouth, the hair, cosmetics, hair dyes and other toilet articles are given, also a detailed description of aromatic substances; their nature, tests of purity, and wholesale manufacture. A book of general, as well as profes- sional interest, meeting the wants not only of the druggist and perfume manufacturer, but also of the general public. Third edition. 312 pages. Illustrated. $3.00 PLUMBING MODERN PLUMBING ILLUSTRATED. By R M. STARBUCK. The author of this book, Mr. R. M. Starbuck, is one of the leading authorities on plumbing in the United States. The book represents the highest standard of plumbing work. It has been adopted and used as a reference book by the United States Government, in its sanitary work in Cuba, Porto Rico and the Philippines, and by the principal Boards of Health of the United States and Canada. It gives Connections, Sizes and Working Data for All Fixtures and Groups of Fixtures. It is helpful to the Master Plumber in Demonstrating to his customers and in figuring work. It gives the Mechanic and Student, quick and easy Access to the best Modern Plumbing Practice. Suggestions for Estimating Plumb- ing Construction are contained in its pages. This book repre- sents, in a word, the latest and best up-to-date practice, and should be in the hands of every architect, sanitary engineer and plumber who wishes to keep himself up to the minute on this important feature of construction. 400 octavo pages, fully illustrated by 55 full- page engravings. 84.00 RUBBER HENLEY'S TWENTIETH CENTURY BOOK OF RE- CEIPTS, FORMULAS AND PROCESSES. Edited by GARD- NER D. Hiscox. Contains upward of 10,000 practical receipts, ...... ., , ..'...,.. S3.00 including among them formulas on artificial rubber. See page 24 for full description of this book. RUBBER HAND STAMPS AND THE MANIPULATION OF INDIA RUBBER. By T. O'CoNOR SLOANE. This book gives full details on all points, treating in a concise and simple manner the elements of nearly everything it is necessary to under- stand for a commencement in any branch of the India Rubber Manufacture. The making of all kinds of Rubber Hand Stamps, Small Articles of India Rubber, U. S. Government Composi- tion, Dating Hand Stamps, the Manipulation of Sheet Rubber, Toy Balloons, India Rubber Solutions, Cements, Blackings, Renovating Varnish, and Treatment for India Rubber Shoes, etc.; the Hektograph Stamp Inks, and Miscellaneous Notes, with a Short Account of the Discovery, Collection, and Manufac- ture of India Rubber are set forth in a manner designed to be readily understood, the explanations being plain and simple. Second edition. 144 pages. Illustrated. $1.00 19 SAWS SAW FILING AND MANAGEMENT OF SAWS. By ROBERT GRIMSHAW. A practical hand book on filing, gumming, swaging, hammering, and the brazing of band saws, the speed, work, and power to run circular saws, etc. A handy book for those who have charge of saws, or for those mechanics who do their own filing, as it deals with the proper shape and pitches of saw teeth of all kinds and gives many useful hints and rules for gumming, setting, and filing, and is a practical aid to those who use saws for any purpose. New edition, revised and enlarged. Illustrated. 81.00 SCREW CUTTING THREADS AND THREAD CUTTING. By COLVIN and STABEL. This clears up many of the mysteries of thread- cutting, such as double and triple threads, internal threads, catch- ing threads, use of hobs, etc. Contains a lot of useful hints and several tables. 25 cents SHEET METAL WORK DIES, THEIR CONSTRUCTION AND USE FOR THE MODERN WORKING OF SHEET METALS. By J. V. WOODWORTH. A new book by a practical man, for those who wish to know the latest practice in the working of sheet metals. It shows how dies are designed, made and used, and those who are engaged in this line of work can secure many valuable suggestions. $3.00 PUNCHES, DIES AND TOOLS FOR MANUFACTUR- ING IN PRESSES. By J. V. WOODWORTH. A work of 5.00 pages and illustrated by nearly 700 engravings, being an en- cyclopedia of die-making, punch-making, die sinking, sheet- metal working, and making of special tools, subpresses, devices and mechanical combinations for punching, cutting, bending, forming, piercing, drawing, compressing, and assembling sheet- metal parts and also articles of other materials in machine tools. $4.00 STEAM ENGINEERING AMERICAN STATIONARY ENGINEERING. By W. E. CRANE. A new book by a well-known author. Begins at the boiler room and takes in the whole power plant. - Contains the result of years of practical experience in all sorts of engine rooms and gives exact information that cannot be found else- where. It's plain enough for practical men and yet of value to those high in the profession. Has a complete examination for a license. 82.00 " BOILER ROOM CHART. By GEO. L. FOWLER. A Chart size 14x28 inches showing in isometric perspective the mechanisms belonging in a modern boiler room. Water tube boilers, ordinary grates and mechanical stokers, feed water heaters and pumps comprise the equipment. The various parts are shown broken or removed, so that the internal construction is fully illustrated. Each part is given a reference number, and these, with the corresponding name, are given in a glossary printed at the sides, 'ihis chart is really a dictionary of the boiler room the names of more than 200 parts being given. It is educational worth many times its cost. 25 cents ENGINE RUNNER'S CATECHISM. By RpBERT GRIM- SHAW. Tells how to erect, adjust, and run the principal steam engines in use in the United States. The work is of a handy size for the pocket. To young engineers this catechism will be of great value, especially to those who may be preparing to go forward to be examined for certificates of competency; and to engineers generally it will be of no little service as they will find in this volume more really practical and useful information than is to be found anywhere else within a like compass. 387 pages. Sixth edition. 83.00 ENGINE TESTS AND BOILER EFFICIENCIES. By J. BUCHETTI. This work fully describes and illustrates the method of testing the power of steam engines, turbine and explosive motors. The properties of steam and the evapora- tive power of fuels. Combustion of fuel and chimney draft; with formulas explained or practically computed. 255 pages, 179 illustrations. $3.00 HORSE POWER CHART. Shows the horse power of any stationary engine without calculation. No matter what the cylinder diameter or stroke; the steam pressure or cut-off; the revolutions, or whether condensing or non-condensing, it's all there. Easy to use, accurate, and saves time and calculations. Especially useful to engineers and designers. 50 cents MODERN STEAM ENGINEERING IN THEORY AND PRACTICE. By GARDNER D. Hiscox. This is a complete and practical work issued for Stationary Engineers and Firemen dealing with the care and management of Boilers, Engines, Pumps, Superheated Steam, Refrigerating Machinery, Dyna- mos, Motors, Elevators, Air Compressors, and all other branches with which the modern Engineer must be familiar. Nearly 200 Questions with their Answers on Steam and Electrical Engineering, likely to be asked by the Examining Board, are included. 487 pages, 405 engravings. S3.00 STEAM ENGINE CATECHISM. By ROBERT GRIMSHAW. This volume of 413 pages is not only a catechism on the question and answer principle; but it contains formulas and worked-out answers for all the Steam problems that appertain to the opera- tion and management of the Steam Engine. Illustrations of various valves and valve gear with their principles of operation are given. 3 4 tables that are indispensable to every engineer and fireman that wishes to be progressive and is ambitious to become master of his calling are within its pages. It is a most valuable instructor in the service of Steam Engineering. Leading en- gineers have recommended it as a valuable educator for the be- ginner as well as a reference book for the engineer. Sixteenth edition. S2.00 STEAM ENGINEER'S ARITHMETIC. By COLVIN- CHENEY. A practical pocket book for the Steam Engineer. Shows how to work the problems of the engine room and shows "why." Tells how to figure horse-power of engines and boilers; area of boilers; has tables of areas and circumferences; steam tables; has a dictionary of engineering terms. Puts you onto all of the little kinks in figuring whatever there is to figure around a power plant. Tells you about the heat unit; absolute zero; adiabatic expansion; duty of engines; factor of safety; and 1,001 other things; and everything is plain and simple not the hardest way to figure, but the easiest. 50 cents 21 STEAM HEATING AND VENTILATION PRACTICAL STEAM, HOT-WATER HEATING AND VENTILATION. By A. G. KING. This book is the standard and latest work published on the subject and has been prepared for the use of all engaged in the business of steam, hot-water heating and ventilation. It is an original and exhaustive work. Tells how to get heating contracts, how to install heating and ventilating apparatus, the best business methods to be used, with "Tricks of the Trade" for shop use. Rules and data for esti- mating radiation and cost and such tables and information as make it an indispensable work for everyone interested in steam, hot -water heating and ventilation. It describes all the principal systems of steam, hot-water, vacuum, vapor and vacuum- vapor heating, together with the new accelerated systems of hot-water circulation, including chapters on up-to-date methods of ventilation and the fan or blower system of heating and venti- lation. You should secure a copy of this book, as each chapter con- tains a mine of practical information. 367 pages, 300 detailed engravings. 83.00 STEAM PIPES STEAM PIPES: THEIR DESIGN AND CONSTRUC- TION. By WM. H. BOOTH. The work is well illustrated in regard to pipe joints, expansion off sets, flexible joints, and self-contained sliding joints for taking up the expansion of long pipes. In fact, the chapters on the flow of Steam and expansion of pipes are most valuable to all steam fitters and users. The pressure strength of pipes and method of hanging them is well treated and illustrated. Valves and by-passes are fully illustrated and described, as are also flange joints and their proper proportions. Exhaust heads and separators. One of the most valuable chapters is that on superheated steam and the saving of steam by insulation with the various kinds of felting and other materials, with comparison tables of the loss of heat in thermal units from naked and felted steam pipes. Contains 187 pages. $2.00 STEEL AMERICAN STEEL WORKER. By E. R. MARKHAM. The standard work on hardening, tempering and annealing steel of all kinds. A practical book for the machinist, tool maker or superintendent. Shows just how to secure best results in any case that comes along. How to make and use furnaces and case harden; how to handle high-speed steel and how to temper for all classes of work. $2.50 HARDENING, TEMPERING, ANNEALING, AND FORGING OF STEEL. By J. V. WOODWORTH. A new book containing special directions for the successful hardening and tempering of all steel tools. Milling cutters, taps, thread dies, reamers, both solid and shell, hollow mills, punches and dies, and all kinds of sheet-metal working tools, shear blades, saws, fine cutlery and metal-cutting tools of all descriptions, as well as for all implements of steel both large and small, the simplest, and most satisfactory hardening and tempering processes are presented. The uses to which the leading brands of steel may be adapted are concisely presented, and their treatment for work- ing under different conditions explained, as are also the special methods for the hardening and tempering of special brands. 320 pages, 250 illustrations. 83.50 HENLEY'S TWENTIETH CENTURY BOOK OF RE- CEIPTS, FORMULAS AND PROCESSES. Edited by GARD- NER D. Hiscox. The most valuable techno-chemical Receipt book published, giving, among other practical receipts, methods of annealing, coloring, tempering, welding, plating, polishing and cleaning steel. See page 24 for full description of this book. $3.00 WATCH MAKING HENLEY'S TWENTIETH CENTURY BOOK OF RE- CEIPTS, FORMULAS AND PROCESSES. Edited by GARDNER D. Hiscox. Contains upwards of 10,000 practical formulas including many watchmakers' formulas. $3.0O WATCHMAKER'S HANDBOOK. By CLAUDIUS SAUNIER. No work issued can compare with this book for clearness and completeness. It contains 498 pages and is intended as a work- shop companion for those engaged in Watchmaking and allied Mechanical Arts. Nearly 250 engravings and fourteen plates are included. $3.00 WIRELESS TELEPHONES WIRELESS TELEPHONES AND HOW THEY WORK. By JAMES ERSKINE-MURRAY. This work is free from elaborate details and aims at giving a clear survey of the way in which Wireless Telephones work. It is intended for amateur workers and for those whose knowledge of Electricity is slight. Chap- ters contained: How We Hear Historical The Conversion of Sound into Electric Waves Wireless Transmission The Pro- duction of Alternating Currents of High Frequency How the Electric Waves are Radiated and Received The Receiving Instruments Detectors Achievements and Expectations Glossary of Technical Words. Cloth. $1.0O Henley's Twentieth Century Book of Recipes, Formulas and Processes Edited by GARDNER D. HISCOX, M.E. Price $3. 00 Cloth Binding $4. 00 Half Morocco Binding Contains over 10,000 Selected Scientific, Chemical, Technological and Practical Recipes and Processes, including Hundreds of So-Called Trade Secrets for Every Business THIS book of 800 pages is the most complete Book of Recipes ever published, giving thousands of recipes for the manufacture of valuable articles forevery-day use. Hints, Helps, Practical Ideas and Secret Processes are revealed within its pages. It covers every branch of the useful arts and tells thousands of ways of making money and is just the book everyone should have at his command. The pages are filled with matters of intense interest and immeasurable practical value to the Photographer, the Perfumer, the Painter, the Manufacturer of Glues, Pastes, Cements and Mucilages, the Physician, the Druggist, the Electrician, the Brewer, the Engineer, the Foundryman, the Machinist, the Potter, the Tanner, the Confectioner, the Chiropodist, the Manufacturer of Chemical Novelties and Toilet Preparations, the Dyer, the Electroplater, the Enameler, the Engraver, the Provisioner, the Glass Worker, the Goldbeater, the Watchmaker and Jeweler, the Ink Manufacturer, the Optician, the Farmer, the Dairy- man, the Paper Maker, the Metal Worker, the Soap Maker, the Veterinary Surgeon, and the Technologist in general. A book to which you may turn with confidence that you will find what you are looking for. A mine of informa- tion up-to-date in every respect. Contains an immense number of formulas that every one ought to have that are not found in any other work. 10781 212441 "